Question

Solve the given initial value problem. y triple prime plus 11 y double prime plus 38 y prime plus 40 y equals 0y′′′+11y′′+38y′+40y=0 y (0 )equals 0y(0)=0​, y prime (0 )equals negative 19y′(0)=−19​, y double prime (0 )equals 117

Answer 1

The linear homogeneous ODE

`y'''+11y''+38y'+40y=0`

has characteristic equation

`r^3+11r^2+38r+40=0`

which factors to

`(r+2)(r+4)(r+5)=0`

and hence has roots at
`r=-2,-4,-5`. So the characteristic solution to the ODE is

`y_c=C_1e^(-2x)+C_2e^(-4x)+C_3e^(-5x)`

Use the given initial conditions to solve for each C.

`y(0)=0\implies C_1+C_2+C_3=0`

`y'(0)=-19\implies -2C_1-4C_2-5C_3=-19`

`y''(0)=4C_1+16C_2+25C_3=117`

`\implies C_1=-9,C_2=8,C_3=1`

so that the particular solution is

`\boxed{y(x)=-9e^(-2x)+8e^(-4x)+e^(-5x)}`