Question

Solve the initial value problem.

2y” - 11y’ + 12y =0 ; where y(0) = 5 and y’(0) = 15

Answer 1

Given, $2y''-11y'+12y=0.$

Auxiliary equation,

$2m^2-11m+12=0\\ (m-4)(m-\frac{3}{2})=0\\ m=4,\frac{3}{2}$

Therefore, general solution is

$y(x)=c_1e^{4x}+c_2e^{\frac{3}{2}x}.\\ y'=4c_1e^{4x}+\frac{3}{2}c_2e^{\frac{3}{2}x}$

Given, the initial condition $y(0)=5 \space and \space y'(0)=-15$ .

$y(0)=c_1+c_2\\ \implies 5=c_1+c_2\\ and\\ y'(0)=4c_1+\frac{3}{2}c_2 \\ \implies -15= 4c_1+\frac{3}{2}c_2$

Solving above two equations, we get

$c_1=-9, c_2=14$

Therefore, particular solution of the given IVP is $y(x)=-9e^{4x}+14e^{\frac{3}{2}x}$ .