ch02

Differential Equations: An Introduction to Modern Methods and Applications

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Unformatted text preview: Chapter 2 Section 2.1 1. (a) 4 2 2 4 y(t) 4 2 2 4 6 8 10 t (b) All solutions seem to converge to an increasing function as t . (c) The integrating factor is ( t ) = e 3 t . Then e 3 t y + 3 e 3 t y = e 3 t ( t + e- 2 t ) = ( e 3 t y ) = te 3 t + e t = e 3 t y = Z ( te 3 t + e t ) dt = 1 3 te 3 t- 1 9 e 3 t + e t + c = y = t 3- 1 9 + e- 2 t + ce- 3 t . We conclude that y is asymptotic to t/ 3- 1 / 9 as t . 2. (a) 4 2 2 4 y(t) 1 0.5 0.5 1 1.5 2 t 1 (b) All slopes eventually become positive, so all solutions will eventually increase without bound. (c) The integrating factor is ( t ) = e- 2 t . Then e- 2 t y- 2 e- 2 t y = e- 2 t ( t 2 e 2 t ) = ( e- 2 t y ) = t 2 = e- 2 t y = Z t 2 dt = t 3 3 + c = y = t 3 3 e 2 t + ce 2 t . We conclude that y increases exponentially as t . 3. (a) 4 2 2 4 y(t) 1 0.5 0.5 1 1.5 2 t (b) All solutions appear to converge to the function y ( t ) = 1. (c) The integrating factor is ( t ) = e t . Therefore, e t y + e t y = t + e t = ( e t y ) = t + e t = e t y = Z ( t + e t ) dt = t 2 2 + e t + c = y = t 2 2 e- t + 1 + ce- t . Therefore, we conclude that y 1 as t . 4. (a) 2 4 2 2 4 y(t) 1 1 2 3 4 5 t (b) The solutions eventually become oscillatory. (c) The integrating factor is ( t ) = t . Therefore, ty + y = 3 t cos(2 t ) = ( ty ) = 3 t cos(2 t ) = ty = Z 3 t cos(2 t ) dt = 3 4 cos(2 t ) + 3 2 t sin(2 t ) + c = y = + 3cos2 t 4 t + 3sin2 t 2 + c t . We conclude that y is asymptotic to (3sin2 t ) / 2 as t . 5. (a) 4 2 2 4 y(t) 1 0.5 0.5 1 1.5 2 t (b) All slopes eventually become positive so all solutions eventually increase without bound. (c) The integrating factor is ( t ) = e- 2 t . Therefore, e- 2 t y- 2 e- 2 t y = 3 e- t = ( e- 2 t y ) = 3 e- t = e- 2 t y = Z 3 e- t dt =- 3 e- t + c = y =- 3 e t + ce 2 t . We conclude that y increases exponentially as t . 3 6. (a) 4 2 2 4 y(t) 4 2 2 4 t (b) For t > 0, all solutions seem to eventually converge to the function y = 0. (c) The integrating factor is ( t ) = t 2 . Therefore, t 2 y + 2 ty = t sin( t ) = ( t 2 y ) = t sin( t ) = t 2 y = Z t sin( t ) dt = sin( t )- t cos( t ) + c = y = sin t- t cos t + c t 2 . We conclude that y 0 as t . 7. (a) 4 2 2 4 y(t) 4 2 2 4 t (b) For t > 0, all solutions seem to eventually converge to the function y = 0. 4 (c) The integrating factor is ( t ) = e t 2 . Therefore, using the techniques shown above, we see that y ( t ) = t 2 e- t 2 + ce- t 2 . We conclude that y 0 as t . 8. (a) 4 2 2 4 y(t) 4 2 2 4 t (b) For t > 0, all solutions seem to eventually converge to the function y = 0....
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ch02 - Chapter 2 Section 2.1 1. (a) 4 2 2 4 y(t) 4 2 2 4 6...

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