Question

completeness and clarity

2. Find a particular solution for the IVP y" - 2y + 2y = 8(t - 5), y(0) = 0, y0) = 1

Answer 1

Here we proceed through laplace transform.

If laplace transform of y +Y(s) then £ {y"}+sY(s) – sy(0) - y0) and for {y'} +sY(s) – y(0) Here take laplace transform of differential equation {{g” - 2y + 2x} = {{(t - 5)} -55 -05 (s2Y(s) – 5* 0) – 1) – 2(sY(s) - 0) +Y(s) = e ::£{s(t – a)} = e Or, Y(s) [s– 2s + 2] - 1 = e-5s 1 e-59 Hence Y(s) + S2 - 2s + 2 $2 – 2s + 2 Now our nest step is to find inverse laplace transform of Y(s) {-1 { + £-1 { $2 – 2s + 2 y(t) = £-+{Y(s)} 1 £1 52 - 2s + 2 52 2s + 2 1 (s – 1)2 + 1 £-1 = e sint