Differential Equations: Homework 3

Transcription

1 Diffrntial Equations: Homwork 3 Alvin Lin January 08 - May 08

2 Sction.3 Ercis Dtrmin whthr th givn quation is sparabl, linar, nithr, or both. Not sparabl and linar. d + sin() y = 0 Ercis 3 Dtrmin whthr th givn quation is sparabl, linar, nithr, or both. Sparabl and non-linar. (t + ) dt = yt y Ercis 5 Dtrmin whthr th givn quation is sparabl, linar, nithr, or both. Not sparabl and non-linar. d dt + t = sin(t) Ercis 7 d y 3 = 0 d y = 3 µ() = d = d y = 3 d y d = 3 d y = + c

3 Ercis 8 Ercis 9 d = y + + d y = + d y µ() = d y = + d = + d d = ln() = y = ln() + c y = ln() + c dr + r tan θ = sc θ dθ µ(θ) = tan θ dθ = ln(sc θ) = sc θ sc() dr dθ + r tan θ sc θ = sc θ sc() dr + r tan θ sc θ dθ = sc θ dθ dθ r sc θ = tan θ + c r = sin θ + c cos θ Ercis 0 + y = 3 d d + y = 4 µ() = d = ln() = + y = d d + y d = d y = + c y = 3 + c 3

4 Ercis Ercis 5 (t + y + ) dt = 0 dt = t + y + dt y = t + t t µ(t) = dt = t dt y t = t t + t dt y t = t t + t dt ( + ) d + y = 0 d + y t = t (t + ) t + c + y = y = t + c t = t + c t + µ() = d + = ln( +) = + + d + + y = + + d + + y d = + d y + = + + c c y = + + Ercis 7 Solv th initial valu problm. d y = y() = d y µ() = d y = d = d y = + c y = + c d = ln() = 4

5 = + c c = y = Ercis 8 Solv th initial valu problm. d + 4y = 0 y(0) = y = d µ() = 4 d = 4 4 d + 4y4 = 3 4 d + 4y4 d = 3 d y 4 = c y = 3 + c = 3 + c 0 0 c = y = Ercis 0 Solv th initial valu problm. d + 3y + = 3 y() = d + 3 y = 3 µ() = 3 d = 3 ln() = 3 3 d + 3 y = d + 3 y d = d y 3 = c y = c 3 = c c = 9 0 y =

6 Sction.6 Ercis Us th mthod discussd undr Brnoulli Equations to solv: Ercis y y d + y = y d d + y y v = y = y d = y y d + y = d + v = d v = d v µ() = d v = d = d v = c y = c Us th mthod discussd undr Brnoulli Equations to solv: d = ln() = d y = y 3 P () = Q() = n = 3 v = y d = + ( n)p ()v = ( n)q() d + v = d µ() = d = d + v = 4 d + v d = 4 d 6 v = c y = 4 + c y 3 d

7 Ercis 3 Us th mthod discussd undr Brnoulli Equations to solv: Ercis 4 y y d = y y d y = y v = y d y y = y y d y = d v = d + v = + v = 4 d d + v d = d = y d µ() = d = ln() = 4 d v = c y = c Us th mthod discussd undr Brnoulli Equations to solv: d + y = 5( )y v = y d = y d y d + y y = 5( ) y y y y d + = 5 0 d + v = 5 0 d v = 0 5 ( ) d µ() = v 0 5 = ( ) d = ln( ) = 7

8 ( ) d v 0 5 d = ( ) v( ) = 5 + c ( ) y = 5 + c d Ercis 5 Us th mthod discussd undr Brnoulli Equations to solv: d dt + t3 + t = 0 d dt + t = v = d dt + 3 t = 3 3 d dt + t = dt + v t = dt v t = dt = 3 t µ(t) = t t dt v t = t dt v ) dt = t t dt v t = ln(t) + c t = ln(t) + c dt = ln(t) = t Ercis 6 Us th mthod discussd undr Brnoulli Equations to solv: d + y = y v = y 3 d y d + y(y ) = y (y ) 3 d + v = + 3v = 3 d µ() = ln 3 d = 3 3 d + 3v3 = = 3y d

9 3 d + 3v3 d = 3 4 d v 3 = c y 3 3 = c Ercis 7 Us th mthod discussd undr Brnoulli Equations to solv: Ercis 8 r dr dθ = r + rθ θ = r θ + r θ dr dθ r θ = r θ v = r dr dθ r r θ = r r θ dθ v θ = θ dθ = r dr dθ dθ + v θ = θ µ(θ) = θ dθ = ln(θ) = θ θ + vθ = dθ θ dθ + vθ dθ = dθ vθ = θ + c θ r = θ + c Us th mthod discussd undr Brnoulli Equations to solv: d + y3 + y = 0 d + y = y3 y 3 v = y d + y 3 y = y 3 y3 d + v = d = y 3 d 9

10 d v = µ() = d = d v = d v d = d v = ( ) + c y = + c If you hav any qustions, commnts, or concrns, plas contact m at alvin@omgimanrd.tch 0