Question

Solve the initial value problem
`(d^(2)y)/(dx^(2)) + 2 (dy)/(dx) + 5y=0`


`y(0)=2`

`y'(0)=2`

Answer 1

The characteristic equation for this ODE is


`r^2+2r+5=0`

which has roots at
`r=-1\pm2i`, so the general solution is


`y_c=(C_1\cos2x+C_2\sin2x)e^(-x)`

Given
`y(0)=2`, we have


`2=(C_1\cos0+C_2\sin0)e^(-0)\implies C_1=2`

and
`y'(0)=2`, we have (upon differentiating
`y_c`)


`{y_c}'=((2C_2-C_1)\cos2x-(2C_1+C_2)\sin2x)e^(-x)`

`2=((2C_2-2)\cos0-(4+C_2)\sin0)e^(-0)`

`2=2C_2-2\implies C_2=2`

So the particular solution is


`y_c=(2\cos2x+2\sin2x)e^(-x)`