Math 3301 Homework Set 6 Solutions 10 Points. = +. The guess for the particular P ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos 2 t : 4D= 2

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1 Mah 0 Homwork S 6 Soluions 0 oins. ( ps) I ll lav i o you o vrify ha y os sin = +. Th guss for h pariular soluion and is drivaivs is blow. Noi ha w ndd o add s ono h las wo rms sin hos ar xaly h omplimnary soluion. Y = ( B + D) os( ) + ( D B) sin ( ) Y = A+ Bos + Dsin Y = D+ B os + B+ D sin lugging his ino h diffrnial quaion and simplifying givs, Sing offiins qual givs, Th gnral soluion is hn, A+ D os B sin = 7 + os os : D= A= sin : B= 0 B= 0 : A= 7 D= 0 = os( ) + sin ( ) 7 + sin ( ) y. ( ps) I ll lav i o you o vrify ha 6 y = +. Th guss for h pariular soluion and is drivaivs is blow. No ha w ndd o add a on h polynomial guss baus h H would b in h omplimnary soluion ohrwis. 7 Y = A + B + C + D + E + F + G + H 6. ( ps) I ll lav i o you o vrify ha y = + Variaion of aramrs is (don forg o divid by h o h g h g( ) ). A pariular soluion is hn, Th gnral soluion is hn, + 7. Th rs of h info w nd for W = = g = ( ) ( ) Y = d + d = + = + = d d y = + +

2 Mah 0 Homwork S 6 Soluions 0 oins 8. ( ps) I ll lav i o you o vrify mos of h soluion work. Th ky quaniis for h problm ar, Th IV is, Th gnral soluion is, m= = L= = k = = ω = = = Applying h iniial ondiions givs, Now, rdu down o a singl osin. u + u = 0 u 0 = = u 0 = = = os + sin u = 8 8 os + 8 sin u 8 R = + = δ = = δ = δ + π = an In his as h firs angl is h orr on and so h singl osin form of h soluion is, = os u No Gradd 6. I ll lav i o you o vrify ha y = +. Th guss for h pariular soluion and is drivaivs is blow. Noi ha w ndd o add a ono h firs rm baus i was par of h omplimnary soluion. Y = A + B + C + D Y = ( A + A) + B + C Y = ( 6A + A) + B lugging his ino h diffrnial quaion and simplifying givs, A 8B + 6B 8C + B + C 8D = Sing offiins qual givs, : 8B= : 6B 8C = : B+ C 8D= 0 : A = 7 Th gnral soluion is hn, A = B = 0 5 C = 8 D = y = + +. I ll lav i o you o vrify ha 8 08 y = +. Th guss for h pariular soluion and is drivaivs is blow. Noi ha w ndd o add ono h sond rm o kp i ou of h omplimnary soluion.

3 Mah 0 Homwork S 6 Soluions 0 oins Y A B Y A B B Y A B B B = + = + + = lugging his ino h diffrnial quaion and simplifying givs, Sing offiins qual givs, Th gnral soluion is hn, Applying h iniial ondiions givs, 0= y 0 = + = A + B = A = : A = : B = B = = + + y y = + = y 0 = + + = 5. I ll lav i o you o vrify ha 7 7 y = os + sin. Th guss for h pariular soluion and is drivaivs is blow. No ha w ndd o add a on h firs porion of h guss baus his is h omplimnary soluion. No ha w won add a ono h las wo porions of h guss sin hy ar missing h xponnial and so ar okay. ( os( ) sin ( )) os( ) sin ( ) 7 Y = A + B + C + D + E + F + G + H 7. I ll lav i o you o vrify ha 5 y = +. Th rs of h info w nd for Variaion of aramrs is, 5 W 5 5 = g 5 6 A pariular soluion is hn, ( 5 6 ) ( 5 6 ) Y = d + d = d d Th gnral soluion is hn, = + = 5 7 = y Applying h iniial ondiions givs, = y( 0) = + + = 0= y 0 =5 = y = I ll b laving i o you o vrify mos of h soluion work. Th ky quaniis for h problm ar,

4 Mah 0 Homwork S 6 Soluions 0 oins ( 0)(.8).5 m= 0 L=.5 k = = 5. γ CR = 0 5. = 68 W hav ovr damping in his as. Th IV is hn, Th gnral soluion is, 0u + 68u + 5.u = 0 u 0 = 0.5 u 0 = 0. u.8.8 = + u =0.5. Applying h iniial ondiions givs : Taking wha w an from #8 w hav h following, 8 8 u + u = 0 u 0 = u 0 = u = os + sin 6 8 Now, baus ω = = ω w do NOT hav rsonan. Hr is som of h Undrmind 0 Coffiins work for his problm. U = Aos + Bsin Aos + Bsin = 5sin 6 6 Sing offiins qual and solving givs A = 0, B = 80. Th gnral soluion is hn, Applying h iniial ondiions givs, 8 8 = os + sin 80sin ( ) u = os 8 + sin 8 80sin ( ) u 8 Now, rdu h firs sin/osin pair down o a singl osin. 8 R = + = δ = = δ = δ + π = an In his as h firs angl is orr. Th aion soluion is hn, = os sin ( ) u. Taking wha w an from # w hav h following, 0u + 68u + 5.u = sin 0 os u 0 = 0.5 u 0 = 0. u = + Undrmind Coffiins will probably b h asis for a pariular soluion so, U = Aos + Bsin.8.8 ( 05.A+ 68B) os + ( 68A 05.B) sin = sin 0 os Sing offiins qual givs A = 0.05, B= Th gnral soluion is hn,.8.8 u = os sin Applying h iniial ondiions givs,.8.8 u 0.60 = os sin Now rdu h sin/osin down o a singl osin.

5 Mah 0 Homwork S 6 Soluions 0 oins 0.05 R = = δ = an = δ = δ + π =.60 In his as h sond angl is orr. Th aual soluion is hn, = os(.60) u