midterm2-B-solution-public

Yp a1 xex yp ax 2ex yp yp 2yp ax

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Unformatted text preview: a particular solution: yp = axe−x (we need x multiplied to e−x because e−x is a solution to the homogeneous part). – – – – ￿ ￿￿ yp = a(1 − x)e−x , yp = a(x − 2)e−x , ￿￿ ￿ yp − yp − 2yp = a(x − 2)e−x − a(1 − x)e−x − 2axe−x = −3ae−x = −6e−x − 3a = −6, a = 2 So, yp = 2xe−x • The general solution is y = yp + c1 y1 + c2 y2 = 2xe−x + c1 e−x + c2 e−x • Alternatively, use variation of parameters – y = u1 y 1 + u2 y 2 , ￿ ￿ – The Wronskian W = y1 y2 − y1 y2 = e−x (2e2x ) + e−x e2x = 3ex y2 f ( x) e 2x ( − 6 e − x ) =− =2 W 3e x y1 f ( x ) e −x ( − 6 e − x ) = = = − 2 e − 3x W 3e x u￿1 = − u￿2 1 – Integrate: u1 = ˆ 1dx = x + c1 u2 = ˆ 2 −2e−3x dx = e−3x + c2 3 – So, the general solution is 2 y1 = 2xe−x + c1 e−x + e−3x e2x + c2 e2x 3 2 −x −x = 2xe + (c1 + )e + c2 e2x 3 = 2xe−x + c3 e−x + c2 e2x where c3 = c1 + 2/3. 3. [5 marks] Find the solution of the initial value problem y ￿ y ￿￿ = x, y (1) = 1, y ￿ (1) = 1 • Let u = y ￿ , y ￿￿ = u￿ , then, uu￿ = x, • separable for u, udu = xdx • Integra...
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