Answer to Question #222277 in Differential Equations for maxie

Question

use the laplace method to solve the equation

d²y/dx²+9y=9x y(0)=1 y'(0)=5

Answer 1

$\text{The given differential equation can be written as }y''+9y-9x=0\\\text{Next, we take the laplace transform of the differential equation}\\L(y'') =S^2L(y)-Sy(0)-y'(0)\\L(y')=SL(y)-y(0) \\L(y)=\overline{y}\\\implies S^2\overline{y}-Sy(0)-y'(0)+9\overline{y}=\frac{9}{s^2} \\\overline{y}(s^2+9)-s-5= \frac{9}{s^2} \\\overline{y}(s^2+9)= \frac{9}{s^2}+s+5=\frac{s^3+5s^2+9}{s^2(s^2+9)}-(1)\\\text{Resolving (1) into partial fractions, where }\\\frac{s^3+5s^2+9}{s^2(s^2+9)}=\frac{A}{S}+\frac{B}{S^2}+\frac{CS+D}{S^2+9 } \\\implies s^3+5s^2+9=AS(S^2+9)+B(S^2+9)+S^2(CS+D) \\\text{Comparing co-efficients, we have that, A=0, B=1, C=1 and D=4} \\\implies \frac{1}{S^2}+\frac{S+4}{S^2+9}\\\text{Taking the inverse Laplace Transform, we have}\\y=x+cos3x+\frac{4}{3}sin3x$