Solve e^ax*y^prime-y*ae^ax=ke^2ax

$$e ^ { a x } \cdot y ^ { \prime } - y \cdot a e ^ { a x } = k e ^ { 2 a x }$$

Solve for k

$$k=-\frac{ay}{e^{ax}}$$

Steps for Solving Linear Equation

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

$$ke^{2ax}=e^{ax}\frac{\mathrm{d}}{\mathrm{d}x}(y)-yae^{ax}$$

Reorder the terms.

$$ke^{2ax}=e^{ax}\frac{\mathrm{d}}{\mathrm{d}x}(y)-aye^{ax}$$

The equation is in standard form.

$$e^{2ax}k=-aye^{ax}$$

Divide both sides by e^{2ax}.

$$\frac{e^{2ax}k}{e^{2ax}}=-\frac{aye^{ax}}{e^{2ax}}$$

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

$$k=-\frac{aye^{ax}}{e^{2ax}}$$

Divide -aye^{ax} by e^{2ax}.

$$k=-\frac{ay}{e^{ax}}$$