What is the solution for -18x^2y^{''}+4x(x+9)y^'-4(x+9)y=-4x^3 ?

The solution for -18x^2y^{''}+4x(x+9)y^'-4(x+9)y=-4x^3 is y=c_{1}e^{(2x)/9}x+c_{2}x-x^2

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-18x^2y^{''}+4x(x+9)y^'-4(x+9)y=-4x^3

- 18 x^{2} y^{^{''}} + 4 x (x + 9) y^{^{'}} - 4 (x + 9) y = - 4 x^{3}

y = c_{1} e^{\frac{2 x}{9}} x + c_{2} x - x^{2}

Solution steps

- 18 x^{2} y^{''} (x) + 4 x (x + 9) y^{'} (x) - 4 (x + 9) y = - 4 x^{3}

Solve using substitution:

y = c_{1} e^{\frac{2 x}{9}} x + c_{2} x - x^{2}

y = c_{1} e^{\frac{2 x}{9}} x + c_{2} x - x^{2}

x \to 9 lim (x)

\int \frac{x}{25 + x ^{2}} d x

x \to 4 lim (e^{4} + 4)

x \to 5 lim (\frac{x}{x ^{2} - 25})

\int \frac{sin ^{2} ( x )}{cos ^{2} ( x )} d x

What is the solution for -18x^2y^{''}+4x(x+9)y^'-4(x+9)y=-4x^3 ?
The solution for -18x^2y^{''}+4x(x+9)y^'-4(x+9)y=-4x^3 is y=c_{1}e^{(2x)/9}x+c_{2}x-x^2