Question

Solve by variation of parameter method `(d^2y)/(dx^2)+3(dy)/(dx)+2y=e^(e^x)`.

Answer 1

`(d^2y)/(dx^2)+3(dy)/(dx)+2y=e^(e^x)`.

Put D=`d/(dx)`   `thereforeD^2y+3Dy+2y=0`

For complementary solution,
f(D)=0
`thereforeD^2+3D+2=0`

𝑫= −𝟏 ,−𝟐

`therefore y_c=c_1e^(x)+c_2e^(-2x)`

Particular integral is given by ,

`y_p=y_1p_1+y_2p_2`

where`p_1=int(-y_2x)/w dx`

`p_1=int(y_1x)/w dx`

`w=|(y_1,y_2),(y'_1,y'_2)|`

`therefore w=|(e^(-x),e^(-2x)),(-e^(-x),-2e^(-2x))|=-e^(-3x)`

`p_1=int(e^(-2x)e^(e^x))/e^(-3x) dx=inte^(e^x)e^xdx=inte^t dt=e^(e^x) ........{"Put"  e^x=t=>e^xdx=dt}`

`p_2=int(e^(-2x))/e^(-3x)e^(e^x) dx=inte^(e^x)e^(2x) dx=int t e^t dt=e^x e^(e^x)-e^(e^x)`

`therefore y_p=e^x e^(e^x)-(e^x e^(e^x)-e^(e^x))e^(-2x)=e^(-2x) e^(e^x)`

The general solution of given differential eqn is given by ,

`y_g=y_c+y_p=c_1e^(-x)+c_2e^(-2x)+e^(-2x)e^(e^x)`