What is the solution for 4(d^2y)/(dx^2)+12(dy)/(dx)+9y=0 ?

The solution for 4(d^2y)/(dx^2)+12(dy)/(dx)+9y=0 is y=c_{1}e^{-(3x)/2}+c_{2}e^{-(3x)/2}x

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4(d^2y)/(dx^2)+12(dy)/(dx)+9y=0

4 \frac{d ^{2} y}{d x ^{2}} + 12 \frac{d y}{d x} + 9 y = 0

y = c_{1} e^{- \frac{3 x}{2}} + c_{2} e^{- \frac{3 x}{2}} x

Solution steps

4 \frac{d ^{2} y}{d x ^{2}} + 12 \frac{d y}{d x} + 9 y = 0

Solve linear ODE:

y = c_{1} e^{- \frac{3 x}{2}} + c_{2} e^{- \frac{3 x}{2}} x

y = c_{1} e^{- \frac{3 x}{2}} + c_{2} e^{- \frac{3 x}{2}} x

y^{^{''}} - 6 y^{^{'}} + 5 y = 0

2 y^{^{''}} + 4 y^{^{'}} + 7 y = 0

x^{^{''}} + 9 x = 0, x^{^{'}} (0) = 4, x (0) = 3

y^{^{''}} + 2 y = 0, y (0) = 2, y^{^{'}} (0) = 22

\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} - 15 y = 0

What is the solution for 4(d^2y)/(dx^2)+12(dy)/(dx)+9y=0 ?
The solution for 4(d^2y)/(dx^2)+12(dy)/(dx)+9y=0 is y=c_{1}e^{-(3x)/2}+c_{2}e^{-(3x)/2}x