What is the solution for 4y^{''}-4y^'+y=32e^{t/2} ?

The solution for 4y^{''}-4y^'+y=32e^{t/2} is y=c_{1}e^{t/2}+c_{2}e^{t/2}t+4e^{t/2}t^2

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4y^{''}-4y^'+y=32e^{t/2}

4 y^{^{''}} - 4 y^{^{'}} + y = 32 e^{\frac{t}{2}}

y = c_{1} e^{\frac{t}{2}} + c_{2} e^{\frac{t}{2}} t + 4 e^{\frac{t}{2}} t^{2}

Solution steps

4 y^{''} (t) - 4 y^{'} (t) + y = 32 e^{\frac{t}{2}}

Solve linear ODE:

y = c_{1} e^{\frac{t}{2}} + c_{2} e^{\frac{t}{2}} t + 4 e^{\frac{t}{2}} t^{2}

y = c_{1} e^{\frac{t}{2}} + c_{2} e^{\frac{t}{2}} t + 4 e^{\frac{t}{2}} t^{2}

y^{^{'''}} + 6 y^{^{''}} + 9 y^{^{'}} = 6 x + 2

y^{^{'''}} - 6 y^{^{''}} + 12 y^{^{'}} - 8 y = t^{3} e^{2 t} + 1

y^{^{''}} + 4 y = e^{3 x} + x sin (2 x)

y^{^{''}} + 3 y^{^{'}} + 2 y = \frac{1}{( 1 + e ^{x} )}

y^{^{'''}} - 9 y^{^{''}} + 108 y = e^{6 x}

What is the solution for 4y^{''}-4y^'+y=32e^{t/2} ?
The solution for 4y^{''}-4y^'+y=32e^{t/2} is y=c_{1}e^{t/2}+c_{2}e^{t/2}t+4e^{t/2}t^2