What is the solution for d^2(d^2+1)y=1 ?

The solution for d^2(d^2+1)y=1 is y=c_{1}+c_{2}x+c_{3}cos(x)+c_{4}sin(x)+(x^2)/2

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d^2(d^2+1)y=1

d^{2} (d^{2} + 1) y = 1

y = c_{1} + c_{2} x + c_{3} cos (x) + c_{4} sin (x) + \frac{x ^{2}}{2}

Solution steps

d^{2} (d^{2} + 1) y = 1

Solve linear ODE:

y = c_{1} + c_{2} x + c_{3} cos (x) + c_{4} sin (x) + \frac{x ^{2}}{2}

y = c_{1} + c_{2} x + c_{3} cos (x) + c_{4} sin (x) + \frac{x ^{2}}{2}

y^{^{'''}} + 3 y^{^{''}} = - 36 x^{2} + 12 x + 42

y^{^{''}} - y^{^{'}} = e^{2 x} \cdot cos (e^{x})

y^{^{''}} + 5 y^{^{'}} + 4 y = 8 x^{2} + 3 + 2 cos^{2} (x)

y^{^{''}} - 3 y^{^{'}} + 10 y = cosh (2 x)

(d^{2} + 4) y = tan^{2} (x)

What is the solution for d^2(d^2+1)y=1 ?
The solution for d^2(d^2+1)y=1 is y=c_{1}+c_{2}x+c_{3}cos(x)+c_{4}sin(x)+(x^2)/2