Solve d^2y/dx^2=x^2e^3x(Ax+B)

$$\frac { d ^ { 2 } y } { d x ^ { 2 } } = x ^ { 2 } e ^ { 3 x } ( A x + B )$$

Solve for A

$$\left\{\begin{matrix}A=-\frac{B}{x}\text{, }&x\neq 0\\A\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x^{2}e^{3x} by Ax+B.

$$\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}=e^{3x}Ax^{3}+x^{2}e^{3x}B$$

Swap sides so that all variable terms are on the left hand side.

$$e^{3x}Ax^{3}+x^{2}e^{3x}B=\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}$$

Subtract x^{2}e^{3x}B from both sides.

$$e^{3x}Ax^{3}=\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}-x^{2}e^{3x}B$$

Reorder the terms.

$$Ax^{3}e^{3x}=\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}-Bx^{2}e^{3x}$$

The equation is in standard form.

$$x^{3}e^{3x}A=-Bx^{2}e^{3x}$$

Divide both sides by x^{3}e^{3x}.

$$\frac{x^{3}e^{3x}A}{x^{3}e^{3x}}=-\frac{Bx^{2}e^{3x}}{x^{3}e^{3x}}$$

Dividing by x^{3}e^{3x} undoes the multiplication by x^{3}e^{3x}.

$$A=-\frac{Bx^{2}e^{3x}}{x^{3}e^{3x}}$$

Divide -Bx^{2}e^{3x} by x^{3}e^{3x}.

$$A=-\frac{B}{x}$$

Solve for B

$$\left\{\begin{matrix}\\B=-Ax\text{, }&\text{unconditionally}\\B\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x^{2}e^{3x} by Ax+B.

$$\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}=e^{3x}Ax^{3}+x^{2}e^{3x}B$$

Swap sides so that all variable terms are on the left hand side.

$$e^{3x}Ax^{3}+x^{2}e^{3x}B=\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}$$

Subtract e^{3x}Ax^{3} from both sides.

$$x^{2}e^{3x}B=\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}-e^{3x}Ax^{3}$$

Reorder the terms.

$$Bx^{2}e^{3x}=\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}-Ax^{3}e^{3x}$$

The equation is in standard form.

$$x^{2}e^{3x}B=-Ax^{3}e^{3x}$$

Divide both sides by x^{2}e^{3x}.

$$\frac{x^{2}e^{3x}B}{x^{2}e^{3x}}=-\frac{Ax^{3}e^{3x}}{x^{2}e^{3x}}$$

Dividing by x^{2}e^{3x} undoes the multiplication by x^{2}e^{3x}.

$$B=-\frac{Ax^{3}e^{3x}}{x^{2}e^{3x}}$$

Divide -Ax^{3}e^{3x} by x^{2}e^{3x}.

$$B=-Ax$$