What is the solution for y^{''}-y=x+sin(x),y(0)=2,y^'(0)=3 ?

The solution for y^{''}-y=x+sin(x),y(0)=2,y^'(0)=3 is y=(2+5/4)e^x-5/4 e^{-x}-x-1/2 sin(x)

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y^{''}-y=x+sin(x),y(0)=2,y^'(0)=3

y^{^{''}} - y = x + sin (x), y (0) = 2, y^{^{'}} (0) = 3

y = (2 + \frac{5}{4}) e^{x} - \frac{5}{4} e^{- x} - x - \frac{1}{2} sin (x)

Solution steps

y^{''} (x) - y = x + sin (x)

Solve linear ODE:

y = (2 + \frac{5}{4}) e^{x} - \frac{5}{4} e^{- x} - x - \frac{1}{2} sin (x)

y = (2 + \frac{5}{4}) e^{x} - \frac{5}{4} e^{- x} - x - \frac{1}{2} sin (x)

y^{^{''}} - 5 y^{^{'}} + 6 y = 20 sin (8 x)

y^{^{''}} + 2 y^{^{'}} + y = cos (x)

y^{^{''}} + 4 y = g (t), y (0) = 0, y^{^{'}} (0) = 0

\frac{d ^{2} y}{d x ^{2}} - 8 \frac{d y}{d x} + 6 y = 3 x^{2} - 8

y^{^{'''}} + 8 y = 2 x - 5 + 8 e^{- 2} x

What is the solution for y^{''}-y=x+sin(x),y(0)=2,y^'(0)=3 ?
The solution for y^{''}-y=x+sin(x),y(0)=2,y^'(0)=3 is y=(2+5/4)e^x-5/4 e^{-x}-x-1/2 sin(x)