Differential Equations Answers

<e> A string of iength L is stretched and fastened to two fix points. Find the solution of

the r.{ave equatiorl (vibrating string) ytt = a^2.yxx, when initial displacernent

y(x,0) = f (x) = b sin (pi.x / t).

also find the Fourier cosine transformation of exp(-x^2)

<e> a) Establish the relation 𝑑

𝑛

𝑑𝑥

𝑛

(𝑥

2 − 1)

𝑛 = 𝑛! 2

𝑛 𝑃𝑛(𝑥)

b) Use Laplace transform to solve

𝑑

2𝑦

𝑑𝑡

2 + 2

𝑑𝑦

𝑑𝑡 + 𝑦 = 𝑡𝑒

−𝑡 𝑤𝑖𝑡ℎ 𝑦(0) = 1, 𝑦

′

(0) = −2

<e>Find a power series solution of 𝑥𝑦′=𝑦 .

determine the general/particular solution for each equation using the applicable solution to equations of order one (separable, homogenous,linear,exact)

x^2 dy/dx + sinx - y = 0

determine the general/particular solution for each equation using the applicable solution to equations of order one (separable, homogenous,linear,exact)

(x^2-xy+y^2)dx-xy dy = 0