What is the solution for y^{''}-2y^'+y=xe^xln(x) ?

The solution for y^{''}-2y^'+y=xe^xln(x) is y=c_{1}e^x+c_{2}xe^x+(6e^xx^3ln(x)-5e^xx^3)/(36)

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y^{''}-2y^'+y=xe^xln(x)

y^{^{''}} - 2 y^{^{'}} + y = x e^{x} ln (x)

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{6 e ^{x} x ^{3} ln ( x ) - 5 e ^{x} x ^{3}}{36}

Solution steps

y^{''} (x) - 2 y^{'} (x) + y = x e^{x} ln (x)

Solve linear ODE:

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{6 e ^{x} x ^{3} ln ( x ) - 5 e ^{x} x ^{3}}{36}

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{6 e ^{x} x ^{3} ln ( x ) - 5 e ^{x} x ^{3}}{36}

x \to 3 lim (3 x^{3} - 5 x^{2} + x - 8)

\int \frac{1}{x ^{2} + 6 x + 5} d x

x \to 4 + lim (\frac{1}{x ^{2} - 16})

\frac{d ^{2}}{d x ^{2}} (\frac{x}{x ^{2} + 4})

\int \frac{64 x ^{2} - 49}{x ^{3}} d x

What is the solution for y^{''}-2y^'+y=xe^xln(x) ?
The solution for y^{''}-2y^'+y=xe^xln(x) is y=c_{1}e^x+c_{2}xe^x+(6e^xx^3ln(x)-5e^xx^3)/(36)