What is the solution for 4(d^2y)/(dx^2)-4(dy)/(dx)+y=0 ?

The solution for 4(d^2y)/(dx^2)-4(dy)/(dx)+y=0 is y=c_{1}e^{x/2}+c_{2}e^{x/2}x

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4(d^2y)/(dx^2)-4(dy)/(dx)+y=0

4 \frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} + y = 0

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

Solution steps

4 \frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} + y = 0

Solve linear ODE:

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

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

41 y - 8 y^{^{'}} + y^{^{''}} = 0, y (0) = - 3, y^{^{'}} (0) = - 32

y^{^{''}} + 4 y = 0, y (0) = 3, y^{^{'}} (0) = 6

y^{^{''}} + 8 y^{^{'}} - 16 y = 0

y^{^{'''}} - 2 y^{^{''}} - 24 y^{^{'}} = 0

\frac{y ^{^{''}} + y ^{^{'}}}{y} = 0

What is the solution for 4(d^2y)/(dx^2)-4(dy)/(dx)+y=0 ?
The solution for 4(d^2y)/(dx^2)-4(dy)/(dx)+y=0 is y=c_{1}e^{x/2}+c_{2}e^{x/2}x