ODES Package. Aleksas Domarkas. November 18, 2013

Transcription

1 ODES Package Aleksas Domarkas November 8, 3 We use open-source computer algebra system(cas) maxima The ODES package contains commands that help you work with ordinary differential equations. List of functions in ODES package: odecv dchange odec solvet ode ic ode ic P iter odetaylor odetaylor odeexact intfactor odel odel ic fs partsol odem odem ic matrix exp odelinsys wronskian A.Domarkas, VU, 3

2 odecv.wxm / odecv Function: odecv(tr,eq,y,x) Makes the change of independed variable in ODE. (%i) load(odes)$. x^3y''' + xy' - y = x (%i) eq:x^3*'diff(y,x,3)+x*'diff(y,x)-y=x$ (%i3) odecv(x=exp(t),eq,y,x); (%o3) d3 y-3 d d t 3 y d t +3 d d t y -y=%et (%i) odel(%,y,t); (%o) y=t %e t C3 + t %e t C + %e t C + t3 %e t (%i5) sol:subst(t=log(x),%); (%o5) y=x log( x) C3 + x log( x) C+x C+ x log( ) 6. ( + x^)y'' + xy' + y = 6 x 3 (%i6) eq:(+x^)*'diff(y,x,)+x*'diff(y,x)+y=$ (%i7) trans(eq):=block( coeff(lhs(eq),'diff(y,x))/coeff(lhs(eq),'diff(y,x,)), t=radcan(integrate(exp(-integrate(%%,x)),x)))$ (%i8) itr:trans(eq); tr:solve(itr,x)[]; (%o8) t=asinh( x) (%o9) x=sinh( t) (%i) odecv(tr,eq,y,x); (%o) d d y+y= t (%i) ode(%,y,t); (%o) y = %k sin( t ) + %k cos( t) (%i) sol:subst(itr,%); (%o) y = %k sin( asinh( x ) + %k cos( asinh( x) A.Domarkas, VU, 3

3 odecv.wxm / 3. y'' - y'/x + x^y = (%i3) eq:'diff(y,x,)-/x*'diff(y,x)+*x^*y=$ (%i) itr:t=x^; tr:x=sqrt(t); (%o) t=x (%o5) x= t (%i6) odecv(t=x^,eq,y,x); (%o6) t d y + ty= d t (%i7) ode(%,y,t); (%o7) y = %k sin( t ) + %k cos( t) (%i8) sol:subst(itr,%); (%o8) y = %k sin( x ) + %k cos( x ). kamke 3.7. x^ y'''-x^3 y'' + x^ y' = (%i9) eq:*x^*'diff(y,x,3)-*x^3*'diff(y,x,)+*x^*'diff(y,x)=$ (%i) odecv(x=exp(t),eq,y,x); (%o) %e t d 3 d y t 3-6 %et d d y t +6 %et d d t y = (%i) eq:%//exp(t),expand; (%o) d3 d y- d t 3 d y t + d d t y =%e-t (%i) odel(eq,y,t); (%o) y=t %e t C3 + %e t C + C + %e-t ( 3 t %e 3 t - ) 7 (%i3) subst(t=log(x),%),expand; (%o3) y=x log( x) C3+x C + C + x log( x) - 36 x (%i) y=collectterms(rhs(%),log(x),x); (%o) y=x log( x) C3 + +x C + C - 36 x (%i5) sol:subst([c=%k,c=%k,c3=%k3-/],%); (%o5) y = %k3 x log( x ) + %k x - 36 x + %k 3 A.Domarkas, VU, 3

4 dchange.wxm / 3 dchange Function: dchange(tr,eq,y,x,new_func,new_var) Makes the change tr:x=f(new_var) of independent variable x. (%i) load(odes)$ load(contrib_ode)$. (-x^)y'' - xy' + n^y = (%i3) eq:(-x^)*diff(y(x),x,)-x*diff(y(x),x)+n^*y(x)=$ (%i) assume(n>)$ (%i5) tr:x=cos(t); itr:t=acos(x); (%o5) x=cos( t) (%o6) t=acos( x) (%i7) dchange(tr,eq,y(x),x,y(t),t)$ eq:trigsimp(%); (%o8) d y( d t ) t +n y( t ) = (%i9) ode(%,y(t),t); (%o9) y( t ) = %k sin( n t ) + %k cos( nt) (%i) sol:dchange(itr,%,y(t),t,y(x),x); (%o) y( x ) = %k sin( n acos( x ) + %k cos( n acos( x). xy'' + y'/-y = (%i) eq:x*diff(y(x),x,)+diff(y(x),x)/-y(x)=$ (%i) tr:x=t^/; itr:solve(%,t)[]; (%o) x= t (%o3) t= x (%i) eq:dchange(tr,eq,y(x),x,y(t),t),ratsimp; (%o) d y( d t) -y( t ) = t (%i5) ode(%,y(t),t); (%o5) y( t ) = %k %e t + %k %e -t (%i6) sol:dchange(itr,%,y(t),t,y(x),x); (%o6) y( x ) = %k %e x + %k %e - x A.Domarkas, VU, 3

5 dchange.wxm / 3 3. x^y'' - xy' + y = x^5*log(x) (%i7) eq:x^*diff(y(x),x,)-*x*diff(y(x),x)+*y(x)=x^5*log(x); (%o7) x d y( x) d x - x d d x y( x ) +y( x ) =x5 log( x) (%i8) tr:x=exp(t); itr:solve(%,t)[]; (%o8) x=%e t (%o9) t=log( ) x (%i) eq:dchange(tr,eq,y(x),x,y(t),t); (%o) d y( d t) -3 d t d t y( t ) +y( t ) =t %e5 t (%i) ode(%,y(t),t); (%o) y( t ) = ( t-7 ) %e5 t + %k %e t + %k %e t (%i) sol:dchange(itr,%,y(t),t,y(x),x); (%o) y( x ) = x5 ( log( x) -7) + %k x + %k x. kamke.8 (%i3) eq:(*x+)^*diff(y(x),x,)-*(*x+)*diff(y(x),x)-*y(x)=3*x+$ (%i) tr:x=(%e^t-)/; itr:solve(tr,t)[]; (%o) x= %et - (%o5) t=log( x+) (%i6) eq:dchange(tr,eq,y(x),x,y(t),t); (%o6) d y( t) d t -8 d d t y( t ) - y( ) t =3( %et - ) + (%i7) ode(eq,y(t),t); (%o7) y( t ) = %k %e 3 t - 9%et - + %k %e -t 96 (%i8) sol:dchange(itr,%,y(t),t,y(x),x); (%o8) y( x ) = %k( x+) 3 + %k x+ - 9( x+) - 96 (%i9) y(x)=map(factor,rhs(sol)); (%o9) y( x ) = %k( x+) 3 + %k 8 x+5 - x A.Domarkas, VU, 3

6 dchange.wxm 3 / 3 5. (%i3) eq:diff(y(x),x,)-diff(y(x),x)+exp(*x)*y(x)=; (%o3) d y( x) - d d x d x y( x ) +%e x y( x ) = (%i3) tr:x=log(t)/; (%o3) x= log( t) (%i3) itr:solve(tr,t)[]; (%o3) t=%e x (%i33) eq:dchange(tr,eq,y(x),x,y(t),t); (%o33) 6 t d y( t ) + t d d t d t y( t ) +t y( ) t = (%i3) eq:subst(y(t)=y,eq); (%o3) 6 t d d y t + t d y +t y= d t (%i35) contrib_ode(eq,y,t); (%o35) [ y=bessel_y, t %k t / 8 + bessel_j, t (%i36) sol:subst(itr,%[]); (%o36) y=bessel_y, %e x %k %e x/ + bessel_j, %e x %k t / 8 ] %k %e x/ 6 A.Domarkas, VU, 3

7 odec.wxm / odec Function: odec(eq,r,x) Solves ODE in respect to expression r. (%i) load(odes)$. Bernoulli differential equation (%i) eq:'diff(y,x)+*y/(x+)=*sqrt(y)/(x+); (%o) d d x y+ y x+ = y x+ (%i3) odec(eq,sqrt(y),x); (%o3) y= x x+ + %c x+ (%i) ode(eq,y,x); (%o) - log( y- ) = log( x+ ) +%c. boj 36. (%i5) eq:x*'diff(y,x,3)-'diff(y,x,)-x*'diff(y,x)+y=-*x^3; (%o5) x d 3 d y x 3 - d d y-x d x y +y=- x3 d x (%i6) odec(eq,'diff(y,x,)+y,x); (%o6) d d y+y= x y-x3 +%c x (%i7) ode(%,y,x); (%o7) y = %k %e x + %k %e -x +x 3 +( 6 -%c) x (%i8) sol:subst(%c=6-%k3,%); (%o8) y = %k %e x + %k %e -x +x 3 + %k3 x 3. sam.35. (%i9) eq:'diff(x,t)=y+z$ eq:'diff(y,t)=x+z$ eq3:'diff(z,t)=x+y$ (%i) odec(eq+eq+eq3,x+y+z,t)$ s:subst(%c=3*c,%); (%o3) z+y+x=3%e t C 7 A.Domarkas, VU, 3

8 odec.wxm / (%i) odec(eq-eq,y-x,t)$ s:subst(%c=3*c,%); (%o5) y-x=3%e -t C (%i6) odec(eq-eq3,z-y,t)$ s3:subst(%c=3*c3,%); (%o7) z-y=3%e -t C3 (%i8) sol:solve([s,s,s3],[x,y,z])[],expand; (%o8) [ x=-%e -t C3 - %e -t C + %e t C, y=-%e -t C3 + %e -t C + %e t C, z= %e -t C3 + %e -t C + %e t C ] Test: (%i9) subst(sol,[eq,eq,eq3])$ ev(%, nouns)$ makelist(rhs(%[k])-lhs(%[k]),k,,3); (%o) [,,]. filipov 65. (%i) eq:'diff(y,x)=sqrt(*x+*y-); (%o) d y= y+ x- d x (%i3) ode(eq,y,x); (%o3) false (%i) load(contrib_ode)$ (%i5) contrib_ode(eq,y,x); Is p positive, negative or zero? p; (%o5) [ log( y+ x- + ) + y+ x- - x+ =%c] (%i6) odec(eq,*y+*x-,x); (%o6) - log( y+ x- + ) + y+ x- + =x+%c 8 A.Domarkas, VU, 3

9 solvet.wxm / solvet Function: solvet(eq,x) Returns rectform solution of polynomial equation. In "casus irreducibilis" give real solutions expressed in trigonometric functions. One version of "solvet" is: (%i) solvet(eq,x):=block([polf,spr,k], spr:solve(eq,x), polf(x):=block([rx], rx:rectform(x), if freeof(%i,x) or atom(x) or freeof(sin,rx) then return(rx) else map(polarform,x), rectform(%%), trigsimp(%%), trigreduce(%%)), makelist(x=polf(rhs(spr[k])),k,,length(spr)), sort(%%) )$ (%i) solvet(x^3-3*x^+,x); (%o) [ x= cos 9 +,x= cos 5 9 +,x= cos ] (%i3) solvet(x^6-3*x^5-3*x^+*x^3-3*x^-6*x+=,x); (%o3) [ x=,x=- 3,x= 3 +,x= cos 9, x= cos 9, x= cos 8 9 ] (%i) solvet(x^3-5*x-5,x); (%o) [ x= 5 cos atan( 9 ) - 3 3, x= 5 cos atan( 9 ) + 3 3, x= 5 cos atan( 9 ) ] 3 (%i5) solve(x^3-5*x-5,x); (%o5) [ x= - 3%i %i + 5 / 3 5 3%i - +, x= 3%i 5 9 %i + 5 / %i + 5 / 3 5-3%i - +, x= 5 9 %i 5 9 %i + 5 / / ] 5 9 %i + 5 / 3 9 A.Domarkas, VU, 3

10 ode_ic.wxm / ode_ic Function: ode_ic(eqn, dvar, ivar, ic) The function ode_ic solves an ordinary differential equation(ode) of first order with initial condition y(x) = y. Here ic is list [x,y]. (%i) load(odes)$. x^y'+3xy = sin(x)/x, y(pi) =. (%i) ode_ic(x^*'diff(y,x) + 3*y*x = sin(x)/x,y,x,[%pi,]); (%o) y=- cos( x ) + x 3. (y^e^y+x)y' = y, y() =. (%i3) eq:(y^*exp(y)+*x)*'diff(y,x)=y$ (%i) ode_ic(eq,y,x,[,]); (%o) ( y 3 -y ) %e y -x = y (%i5) solve(%,x); (%o5) [ x=( y 3 -y )%e y ] 3. xy' + y = y^log(x), y()=/. (%i6) eq:x*'diff(y,x)+y=*y^*log(x)$ (%i7) ode_ic(eq,y,x,[,/]); (%o7) y= log( x ) +. (x^-)y' + xy^ =, y() =. (%i8) eq:(x^-)*'diff(y,x)+*x*y^=$ (%i9) ode_ic(eq,y,x,[,]); (%o9) y= log( -x ) + A.Domarkas, VU, 3

11 ode_ic.wxm / ode_ic Function: ode_ic(eqn, dvar, ivar, ic) The function ode_ic solve an ordinary differential equation(ode) of second order with initial conditions y(x) = y, y'(x) = y. Here ic is list [x, y, y]. (%i) load(odes)$. y'' + yy'^3=, y()=, y'()= (%i) eq:'diff(y,x,) + y*'diff(y,x)^3 = $ (%i3) sol:ode_ic(eq,y,x,[,,]); (%o3) y=( 9 x x) / 3 - ( 9 x x) / 3 Test: (%i) ev(rhs(sol),x=); (%o) (%i5) diff(rhs(sol),x)$ ev(%,x=); (%o6) (%i7) subst(sol,eq)$ ev(%, nouns)$ radcan(%); (%o9) =. y'' = 8*y^3, y() =, y'() = 8. (%i) eq:'diff(y,x,)=8*y^3$ (%i) ode_ic(eq,y,x,[,,8]); (%o) y=- 8 x- 3. y'' + y = /cos(x), y()=, y'()=. (%i) eq:'diff(y,x,)+y = /cos(x)$ (%i3) sol:ode_ic(eq,y,x,[,,]); (%o3) y=cos( x) log( cos( x ) +x sin( x ) +cos( x) A.Domarkas, VU, 3

12 P_iter.wxm / P_iter Function: P_iter(eq, x, y, x, y, n). Solves first order differential equation using Picard iterative process. (%i) load(odes)$. y'=x^+y^, y()= (%i) eq:'diff(y,x)=x^+y^$ (%i) x:$ y:$ (%i) for k: thru 3 do print(p_iter(eq,x,y,x,y,k))$ x 3 3 x x3 3 x x 79 + x x3 3. y' = x( + y), y() =. (%i5) eq:'diff(y,x)=*x*(+y)$ (%i6) x:$ y:$ (%i8) for k: thru 5 do print(p_iter(eq,x,y,x,y,k))$ x x +x x x +x x 8 + x6 6 + x +x x + x8 + x6 6 + x +x A.Domarkas, VU, 3

13 odetaylor.wxm / odetaylor Function: odetaylor(eq, x, y, n). Solves first order differential equation using Taylor-series expansion. (%i) load(odes)$. y'=x+y^, y()= (%i) eq:diff(y(x),x)=x+y(x)^; (%o) d d x y( x ) =y( x) +x (%i3) odetaylor(eq,,,6); (%o3)/t/ +x+ 3 x + x3 7 x 3 x5 9 x y'=x^+y^, y()= (%i) eq:diff(y(x),x)=x^+y(x)^; (%o) d d x y( x ) =y( x) +x (%i5) odetaylor(eq,,,5); (%o5)/t/ x3 3 + x x 3 x y'=x-y^, y()=- (%i6) eq:diff(y(x),x)=x-y(x)^; (%o6) d d x y( x ) =x-y( x) (%i7) odetaylor(eq,,-,5); (%o7)/t/ - + ( x-) + ( x-) 3 + ( x-) + ( x-) A.Domarkas, VU, 3

14 odetaylor.wxm / odetaylor Function: odetaylor(eq, x, y, y, n). Solves second order differential equation using Taylor-series expansion. (%i) load(odes)$. Airy's Equation y''-xy=, y()=, y'()=. (%i) eq:'diff(y(x),x,)-x*y(x)=; (%o) d y( x ) -x y( x ) = d x (%i3) odetaylor(eq,,,,5); (%o3)/t/ + x3 6 + x6 8 + x x 77 + x y''=(y')^+xy, y()=, y'()= (%i) eq:'diff(y(x),x,)='diff(y(x),x,)^+x*y(x); (%o) d y( d x ) = d x d x y( x) +x y( x) (%i5) odetaylor(eq,,,,5); (%o5)/t/ + ( x-) + ( x-) 3 + ( x-) + ( x-) (%i6) eq:'diff(y(x),x,)+x*'diff(y(x),x)+y(x)=; (%o6) d y( x ) +x d d x d x y( x ) +y( x ) = (%i7) odetaylor(eq,,,,5); (%o7)/t/ x- x3 3 + x5 5 - x7 5 + x x x x (%i8) sum((-)^n*^n*n!*x^(*n+)/(*n+)!,n,,7); (%o8) - x x x x x7 5 + x5 5 - x3 3 +x A.Domarkas, VU, 3

15 odeexact.wxm / odeexact (%i) load(odes)$ Function: odeexact(eq). Solves first order exact equation. (%i) eq:*x*y*dx+(x^+3*y^)*dy=; (%o) dy( 3 y +x ) + dx x y = (%i3) odeexact(eq); (%o3) y 3 +x y=c. (%i) eq:(6*x^-y+3)*dx+(3*y^-x-3)*dy=; (%o) dy( 3 y -x-3) +dx( -y+6 x + 3 ) = (%i5) odeexact(eq); (%o5) y 3 -x y-3 y+ x x=c 3. (%i6) eq:exp(y)*dx+(*y+x*exp(y))*dy=; (%o6) dy( x %e y + y ) +dx %e y = (%i7) odeexact(eq); (%o7) x %e y +y =C. (%i8) eq:(x*dx+y*dy)/sqrt(x^+y^)+(x*dy-y*dx)/x^=; dy y + dx x dy x - dx y (%o8) + = y +x x (%i9) odeexact(eq); (%o9) y +x + y x =C 5 A.Domarkas, VU, 3

16 intfactor.wxm / intfactor Function: intfactor(eq, omega). Find intfactor mu = mu(omega) of the first order differential equation. (%i) load(odes)$. (%i) eq:(+y^)*dx+x*y*dy=$ (%i3) intfactor(eq,x); (%o3) x (%i) odeexact(eq*%); (%o) x y + x =C. (%i5) eq:(x*y^-*y^3)*dx+(3-*x*y^)*dy=$ (%i6) intfactor(eq,y); (%o6) y (%i7) odeexact(eq*%); (%o7) - xy- 3 y + x =C 3. (%i8) eq:y*dx+(x^+y^-x)*dy=$ (%i9) odeexact(eq); (%o9) false (%i) intfactor(eq,x^+y^); (%o) y +x (%i) odeexact(eq*%); (%o) y+atan y 6 A.Domarkas, VU, 3

17 intfactor.wxm /. (%i) eq:x*y*dx+(*x^+3*y^-)*dy=$ (%i3) intfactor(eq,y); (%o3) y 3 (%i) odeexact(eq*%); (%o) y 6 + x y - 5 y =C 5. (%i5) eq:(x^*y^3+6*y^5)*dx+(*x^3*y^+*x^)*dy=$ (%i6) intfactor(eq/y,x*y); (%o6) x y (%i7) odeexact(-eq/y*%); (%o7) xy + 3 y + x 3 =C Other method: (%i8) mu:x^a*y^b; (%o8) x a y b (%i9) diff(mu*(x^*y^3+6*y^5),y)=diff(mu*(*x^3*y^+*x^),x)$ (%i) factor(lhs(%)-rhs(%)); (%o) x a y b ( 6 by + 3 y +b x y - ax y - 3 x y - ax 3-8 x 3 ) (%i) collectterms(%/mu,x,y); (%o) ( 6 b+3) y +( b- a-3) x y +( - a-8) x 3 (%i) solve([coeff(%,y^),coeff(%,x^*y^),coeff(%,x^3)],[a,b]); solve: dependent equations eliminated: () (%o) [[a=-,b=-5]] (%i3) 'mu=subst(%[],mu); (%o3) = x y 5 7 A.Domarkas, VU, 3

18 odel.wxm / odel Function: odel(eqn, dvar, ivar) The function odel solves an linear ODEs with constant coefficients. (%i) load(odes)$ load(contrib_ode)$. y''' - y'' + y' =. (%i3) eq:'diff(y,x,3)-*'diff(y,x,)+'diff(y,x) = $ (%i) odel(eq,y,x); (%o) y=x %e x C3 + %e x C + C. y'''' + 8y'' + 6y = x exp(3x) + sin(x)^ + (%i5) eq:'diff(y,x,)+8*'diff(y,x,)+6*y=x*exp(3*x)+sin(x)^+$ (%i6) sol:odel(eq,y,x); (%o6) y=x sin( x) C+sin( x) C3+x cos( x) C+cos( x) C+ 97 x cos( x ) +( 83 x-768 )%e 3 x (%i7) ode_check(eq,sol); (%o7) 3. y'''- 3y'' + y = sin(x)^3. (%i8) eq:'diff(y,x,3)-3*'diff(y,x,)+y=sin(x)^3; (%o8) d3 y-3 d d x 3 y d x +y=sin( x) 3 (%i9) solvet(k^3-3*k^+=,k); (%o9) [ k= cos 5 9 +,k= cos 7 9 +,k= cos 9 + ] (%i) sol:odel(eq,y,x); (%o) y=%e cos 7 x+x cos 5 x+x cos 9 C3 + %e 9 C + %e 9 x+x C - 8 sin( 3 x ) +7 cos( 3 x) -68 sin( x) -67 cos( x) 65 (%i) ode_check(eq,sol); (%o) 8 A.Domarkas, VU, 3

19 odel_ic.wxm / odel_ic Function: odel_ic(eqn, dvar, ivar, ic) The function odel_ic solves initial value problems for linear ODEs with constant coefficients. (%i) load(odes)$ load(contrib_ode)$. y''' + y'' = x + exp(-x), y() =, y'() =, y''() =. (%i3) eq:'diff(y,x,3)+'diff(y,x,)=x + exp(-x); (%o3) d3 d y+ d x3 d x y=%e-x +x (%i) odel_ic(eq, y, x, [,,, ]),expand; (%o) y=x %e -x + %e -x + x3 6 - x + 3 x-3. y''''-y=8*exp(x), y()=, y'()=, y''()=, y'''()=6. (%i5) eq:'diff(y,x,)-y=8*exp(x); (%o5) d d x y-y=8%ex (%i6) odel_ic(eq, y, x, [,,,, 6]); (%o6) y= x %e x 3. y''''' - y' =, y()=, y'()=, y''()=, y'''()=, y''''()=. (%i7) eq:'diff(y,x,5)-'diff(y,x)=; (%o7) d5 d y- d x 5 d x y= (%i8) sol:odel_ic(eq,y,x,[,,,,,]); (%o8) y=cos( x ) +%e x - Test: (%i9) ode_check(eq,sol); (%o9) (%i) makelist(diff(rhs(sol),x,k),k,,)$ ev(%,x=); (%o) [,,,,] 9 A.Domarkas, VU, 3

20 fs.wxm / fs Function: fs(eq, y, x). Find fundamental system of solutions of the linear n-th order differential equation with constant coeficients. (%i) load(odes)$. (%i) eq:'diff(y,x,3)+3*'diff(y,x,)-*'diff(y,x)=x-; (%o) d3 y+3 d d x 3 y d x - d d x y =x- (%i3) fs(eq,y,x); (%o3) [,%e - 5 x,%e x ] (%i) odel(eq,y,x); (%o) y=%e x C3 + %e - 5 x C + C - 5 x - 7 x. (%i5) eq:'diff(y,x,)+8*'diff(y,x,)+6*y=x^*exp(x)*sin(x); (%o5) d d y+8 d x d y x +6 y=x %e x sin( x) (%i6) fs(eq,y,x); (%o6) [ cos( x ), x cos( x ), sin( x ), x sin( x )] (%i7) sol:odel(eq,y,x); (%o7) y=x sin( x) C+sin( x) C3+x cos( x) C+cos( x) C+ ( 75 x - 6 x+68 )%e x sin( x ) +( - x + 8 x+6 )%e x cos( x) 5 3. (%i8) eq:'diff(y,x,8)+'diff(y,x,)=x^5$ (%i9) solvet(k^8+k^=,k); (%o9) [ k=%i, k= %i - 3, k=-%i - 3 3, k=-%i, k= - %i, k=%i + 3, k=] A.Domarkas, VU, 3

21 fs.wxm / (%i) fs(eq,y,x); (%o) [,x,%e - cos( x ), sin( x )] 3 x cos x,%e 3 x cos x,%e- 3 x sin x,%e 3 x sin x, (%i) sol:odel(eq,y,x); (%o) y=sin( x) C8+cos( x) C7+%e cos x C + %e- 3 x 3 x sin x C6 + %e- cos x C3 + x C + C + x7-5 x 3 x sin x C5 + %e 3 x Test: (%i) load(contrib_ode)$ (%i3) ode_check(eq,sol); (%o3). (%i) eq:'diff(y,x,6)-3*'diff(y,x,5)-3*'diff(y,x,)+*'diff(y,x,3) -3*'diff(y,x,)-6*'diff(y,x,)+*y=*x^7+sin(x)^3; (%o) d6 d y-3 d 5 x 6 d y x 5-3 d d y x + d 3 d y x 3-3 d d y x -6 d y + y=sin( ) d x x 3 + x 7 (%i5) fs(eq,y,x); (%o5) [%e x,%e cos 9 (%i6) sol:odel(eq,y,x)$ x cos,%e 9 x cos,%e 8 9 x,%e x- 3 x,%e 3 x+x ] (%i7) expand(%); (%o7) y=%e 3 x+x C6 + %e x- 3 x C5 + %e cos 8 x cos x 9 C + %e 9 C3 + %e cos x 9 C + %e x 93 sin( ) C + 3 x 7 cos( - 3 x ) + 3 sin( x) 63 cos( + x ) +x 7 + x x x x x x+735 Test: (%i8) ode_check(eq,sol)$ trigreduce(%)$ trigrat(%); (%o) A.Domarkas, VU, 3

22 partsol.wxm / partsol Function: partsol(eq, y, x). Find partial solution of the linear n-th order differential equation with constant coefficients. (%i) load(odes)$. (%i) eq:'diff(y,x,3)+3*'diff(y,x,)-*'diff(y,x)=x-; (%o) d3 y+3 d d x 3 y d x - d d x y =x- (%i3) partsol(eq,y,x); (%o3) - 5 x - 7 x (%i) odel(eq,y,x); (%o) y=%e x C3 + %e - 5 x C + C - 5 x - 7 x. (%i5) eq:'diff(y,x,)+'diff(y,x,3)-3*'diff(y,x,)-5*'diff(y,x)-*y= exp(*x)-exp(-x); (%o5) d d3 y+ d x d y-3 d x 3 d y x -5 d d x y - y=%e x - %e -x (%i6) partsol(eq,y,x); (%o6) %e-x ( x %e 3 x + 3 x 3 + x + x) 5 (%i7) odel(eq,y,x); (%o7) y=%e x C + x %e -x C3 + x %e -x C + %e -x C + %e -x ( x %e 3 x + 3 x 3 + x + x) 5 (%i8) expand(%)$ (%i9) y=collectterms(rhs(%),exp(-x),exp(*x)); (%o9) y=%e x C + x 7 + %e-x x C3 + x C + C + x3 8 + x 7 + x 7 A.Domarkas, VU, 3

23 partsol.wxm / 3. (%i) eq:'diff(y,x,3)+'diff(y,x,)=/cos(x)$ (%i) partsol(eq,y,x); log sin( x) - sin( x ) + - sin( x) log( cos( x ) + x cos( x ) (%o) - (%i) sol:odel(eq,y,x); log sin( x) - sin( x ) + - sin( x) log( cos( x ) + x cos( x ) (%o) y=sin( x) C3+cos( x) C + C - Test: (%i3) load(contrib_ode)$ (%i) ode_check(eq,sol); (%o). (%i5) eq:'diff(y,x,3)+8*'diff(y,x,)+9*y=cos(x)^3; (%o5) d3 d y+8 d x 3 y +9 y=cos( ) d x x 3 (%i6) partsol(eq,y,x); 3 sin( 3 x) -39 cos( 3 x) -63 sin( x) -8 cos( ) (%o6) - x 56 (%i7) sol:odel(eq,y,x); (%o7) y=%e x/ sin 5 7 x C3 + %e x/ cos 3 sin( 3 x) -39 cos( 3 x) -63 sin( x) -8 cos( x) x C + %e -x C - Test: (%i8) load(contrib_ode)$ (%i9) ode_check(eq,sol); (%o9) 3 A.Domarkas, VU, 3

24 odem.wxm / odem Function: odem(a,f,t) Find solutions of linear system of ODEs with constant coefficients in matrix form: Y' = AY + F (%i) load(odes)$. Y' = AY + F. (%i) A:matrix([,],[,]); (%o) (%i3) F:transpose([t-,*t-]); (%o3) t- t- (%i) sol:odem(a,f,t); (%o) Test: %e 3 t %e 3 t - %e-t C + + %e-t %e 3 t + %e-t C - t C +( %e 3 t - %e -t ) C + (%i5) diff(sol,t)-a.sol-f$ expand(%); (%o6). Y' = AY. (%i7) A:matrix([,,-8,-3],[-8,-,,],[-9,-3,-5,-9],[33,,9,3]); (%o7) (%i8) F:transpose([,,,])$ A.Domarkas, VU, 3

25 odem.wxm / (%i9) charpoly(a, x),factor; (%o9) ( x - x+3) (%i) solve(%); (%o) [ x= - 3%i,x=3%i+ ] (%i) odem(a,f,t)$ (%i) sol:ratsimp(%); (%o) (- 3 t- )%e t sin( 3 t) C+( (- 9 t-3 )%e t sin( 3 t ) +t %e t cos( 3 t) C3- ( 9 t+3 )%e t sin( 3 t) -9 t %e t cos( 3 t) C+( ( t+ )%e t sin( 3 t) -3 t %e t cos( 3 t) C3+(3 t % - 3%e t sin( 3 t) C+( %e t cos( 3 t) -9%e t sin( 3 t) C ( 9%e t sin( 3 t ) +( 3 t+ )%e t cos( 3 t) C+( ( t+7 )%e t sin( 3 t ) +9 t %e t cos( 3 t) C3+(3 (%i3) sol[,]; (%o3) (- 3 t- )%e t sin( 3 t) C+( (- 9 t-3 )%e t sin( 3 t ) +t %e t cos( 3 t) C3- t %e t sin( 3 t) C+( %e t cos( 3 t) -3 t %e t sin( 3 t) C (%i) sol[,]; (%o) ( 9 t+3 )%e t sin( 3 t) -9 t %e t cos( 3 t) C+ ( t+ )%e t sin( 3 t) -3 t %e t cos( 3 t) C3+ ( 3 t %e t sin( 3 t ) +( - 3 t )%e t cos( 3 t) C+ ( 9 t-3 )%e t sin( 3 t) -9 t %e t cos( 3 t) C (%i5) sol[3,]; (%o5) - 3%e t sin( 3 t) C+( %e t cos( 3 t) -9%e t sin( 3 t) C3-%e t sin( 3 t) C - 3%e t sin( 3 t) C (%i6) sol[,]; (%o6) ( 9%e t sin( 3 t ) +( 3 t+ )%e t cos( 3 t) C+ ( t+7 )%e t sin( 3 t ) +9 t %e t cos( 3 t) C3+( 3%e t sin( 3 t ) +t %e t cos( 3 t) C +( %e t sin( 3 t ) +3 t %e t cos( 3 t) C Test: (%i7) diff(sol,t)-a.sol$ expand(%); (%o8) 5 A.Domarkas, VU, 3

26 odem_ic.wxm / odem_ic Function: odem_ic(a, F, t, t, Y) Find solutions of initial problem for linear system of ODEs in matrix form: Y' = AY + F, Y(t) = Y. (updated version of odelinsys) (%i) load(odes)$. Y' = AY + F, Y()=Y (%i) A:matrix([,-],[,-]); - (%o) - (%i3) F:transpose([*%e^(-*t),]); (%o3) %e-t (%i) Y:transpose([,]); (%o) (%i5) sol:odem_ic(a,f,t,,y); sin( t) (%o5) sin( t) -cos( t ) +%e - t Test: (%i6) diff(sol,t)-a.sol-f$ expand(%); (%o7) (%i8) ev(sol,t=); (%o8) 6 A.Domarkas, VU, 3

27 odem_ic.wxm /. Y' = AY, Y()=transpose([5,35,55,75]). (%i9) A:matrix([,,,7],[,,,],[,,,],[7,,,]); 7 (%o9) 7 (%i) F:transpose([,,,])$ (%i) Y:transpose([5,35,55,75]); 5 (%o) (%i) charpoly(a, x),factor; (%o) ( x-5 )( x- )( x+3( ) x+6) (%i3) sol:odem_ic(a,f,t,,y); 7 %e 5 t + 8 %e t - 3 %e - 3 t (%o3) 5 %e 5 t - 9%e t - %e - 6 t 5 %e 5 t - 9%e t + %e - 6 t 7 %e 5 t + 8 %e t + 3 %e - 3 t Test: (%i) diff(sol,t)-a.sol$ expand(%); (%o5) (%i6) ev(sol,t=); 5 (%o6) A.Domarkas, VU, 3

28 matrix_exp.wxm / matrix_exp Function: matrix_exp(a,t) Returns matrix exponential e^(at) computed via Laplace transforms. (%i) matrix_exp(a,r):= block([n,b,s,t,lap,f], n:length(a), B:invert(s*ident(n)-A), Lap(f):=ilt(f, s, t), matrixmap(lap,b), subst(t=r,%%))$. (%i) A:matrix([,],[,]); (%o) (%i3) matrix_exp(a,t); (%o3) %et t %e t %e t (%i) e^'a=matrix_exp(a,); (%o) e A = %e %e %e. (%i5) A:matrix([,7,6],[-5,-,-6],[,,6]); (%o5) (%i6) e^'a=matrix_exp(a,); (%o6) e A = 3 %e 6 %e - %e - 9%e6 3 %e 6 5%e - 5%e - 9%e6 %e 6 - %e %e - %e6 %e 6 %e 6 %e 6 8 A.Domarkas, VU, 3

29 odelinsys.wxm / odelinsys Function: odelinsys(a, F, x, x, Y) Find solutions of initial problem for linear system of ODEs in matrix form: Y' = AY + F, Y(x) = Y. (%i) load(odes)$ load(diag)$. Solve Y' = AY + F, Y() = Y (%i3) A:matrix([,3],[-,5]); (%o3) 3-5 (%i) F:transpose([-x,*x])$ Y:transpose([3,])$ (%i6) sol:odelinsys(a,f,x,,y); (%o6) 7%e x 5 %e x + + x %e x 3 + 5%ex - x Test: (%i7) diff(sol,x)-a.sol-f,expand; (%o7). Solve Y' = AY (%i8) A:matrix([,-,],[3,,-],[,,]); (%o8) (%i9) sol:odelinsys(a,[,,],t,,[c,c,c3]),factor; (%o9) %e t ( t C3 - t C - tc+t C + tc+ C) %e t ( t C3 - t C3 - t C - t C + C + t C + 3 tc) %e t ( t C3 - tc3+ C3 - t C + t C + tc) 9 A.Domarkas, VU, 3

30 wronskian.wxm / wronskian Function: wronskian ([f_,..., f_n], x) Returns the Wronskian matrix of the list of expressions [f_,..., f_n] in the variable x. (%i) load(odes)$. (%i) wronskian([f(x),g(x),h(x)],x); f( x) g( x) h( x) (%o) d d x f( x) d d x g( x) d d x h( x) d f( x) d x d g( x) d x d h( x) d x. Form a linear homogeneous differential equation, knowing its fundamental system of solutions: y=x, y=x^3. (%i3) depends(y,x); (%o3) [y( ) x ] (%i) wronskian([x,x^3,y],x); x x 3 y (%o) 3 x d d x y 6 x d d x y (%i5) determinant(%)=; (%o5) x 3 x d y d x -6 x d d x y -x3 d y +6 xy= d x (%i6) eq:expand(%/x/); (%o6) x d y d x -3 x d y +3 y= d x (%i7) ode(eq,y,x); (%o7) y = %k x 3 + %k x 3 A.Domarkas, VU, 3

31 References: Kamke, E., 9. Differentialgleichungen. Losungsmethoden und Losungen. AkademischeVerlagsgesellschaft, Leipzig. 3 A.Domarkas, VU, 3