Solve dy/dx=2Ae^2x-2Be^-2x

$$\frac { d y } { d x } = 2 A e ^ { 2 x } - 2 B e ^ { - 2 x }$$

Solve for A

$$A=\frac{B}{e^{4x}}$$

Steps for Solving Linear Equation

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

$$2Ae^{2x}-2Be^{-2x}=\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Add 2Be^{-2x} to both sides.

$$2Ae^{2x}=\frac{\mathrm{d}}{\mathrm{d}x}(y)+2Be^{-2x}$$

The equation is in standard form.

$$2e^{2x}A=\frac{2B}{e^{2x}}$$

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

$$\frac{2e^{2x}A}{2e^{2x}}=\frac{2B}{e^{2x}\times 2e^{2x}}$$

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

$$A=\frac{2B}{e^{2x}\times 2e^{2x}}$$

Divide \frac{2B}{e^{2x}} by 2e^{2x}.

$$A=\frac{B}{e^{4x}}$$

Solve for B

$$B=Ae^{4x}$$

Steps for Solving Linear Equation

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

$$2Ae^{2x}-2Be^{-2x}=\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Subtract 2Ae^{2x} from both sides.

$$-2Be^{-2x}=\frac{\mathrm{d}}{\mathrm{d}x}(y)-2Ae^{2x}$$

The equation is in standard form.

$$\left(-\frac{2}{e^{2x}}\right)B=-2Ae^{2x}$$

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

$$\frac{\left(-\frac{2}{e^{2x}}\right)B}{-\frac{2}{e^{2x}}}=-\frac{2Ae^{2x}}{-\frac{2}{e^{2x}}}$$

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

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

Divide -2Ae^{2x} by -2e^{-2x}.

$$B=Ae^{4x}$$