Solve ax^2+by^prime=c

$$a x ^ { 2 } + b y ^ { \prime } = c$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{c}{x^{2}}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&c=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract b\frac{\mathrm{d}}{\mathrm{d}x}(y) from both sides.

$$ax^{2}=c-b\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Reorder the terms.

$$ax^{2}=-b\frac{\mathrm{d}}{\mathrm{d}x}(y)+c$$

The equation is in standard form.

$$x^{2}a=c$$

Divide both sides by x^{2}.

$$\frac{x^{2}a}{x^{2}}=\frac{c}{x^{2}}$$

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

$$a=\frac{c}{x^{2}}$$

Solve for b (complex solution)

$$b\in \mathrm{C}$$

$$c=ax^{2}$$

Steps for Solving Linear Equation

Subtract ax^{2} from both sides.

$$b\frac{\mathrm{d}}{\mathrm{d}x}(y)=c-ax^{2}$$

Reorder the terms.

$$b\frac{\mathrm{d}}{\mathrm{d}x}(y)=-ax^{2}+c$$

The equation is in standard form.

$$0=c-ax^{2}$$

This is false for any b.

$$b\in$$

Solve for a

$$\left\{\begin{matrix}a=\frac{c}{x^{2}}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&c=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract b\frac{\mathrm{d}}{\mathrm{d}x}(y) from both sides.

$$ax^{2}=c-b\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Reorder the terms.

$$ax^{2}=-b\frac{\mathrm{d}}{\mathrm{d}x}(y)+c$$

The equation is in standard form.

$$x^{2}a=c$$

Divide both sides by x^{2}.

$$\frac{x^{2}a}{x^{2}}=\frac{c}{x^{2}}$$

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

$$a=\frac{c}{x^{2}}$$

Solve for b

$$b\in \mathrm{R}$$

$$c=ax^{2}$$

Steps for Solving Linear Equation

Subtract ax^{2} from both sides.

$$b\frac{\mathrm{d}}{\mathrm{d}x}(y)=c-ax^{2}$$

Reorder the terms.

$$b\frac{\mathrm{d}}{\mathrm{d}x}(y)=-ax^{2}+c$$

The equation is in standard form.

$$0=c-ax^{2}$$

This is false for any b.

$$b\in$$