Solve dy/dx=ae^-bx(c-dy)

$$\frac { d y } { d x } = a e ^ { - b x } ( c - d y )$$

Solve for a

$$\left\{\begin{matrix}\\a=0\text{, }&\text{unconditionally}\\a\in \mathrm{R}\text{, }&c=dy\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply ae^{\left(-b\right)x} by c-dy.

$$\frac{\mathrm{d}(y)}{\mathrm{d}x}=ae^{\left(-b\right)x}c-ae^{\left(-b\right)x}dy$$

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

$$ae^{\left(-b\right)x}c-ae^{\left(-b\right)x}dy=\frac{\mathrm{d}(y)}{\mathrm{d}x}$$

Reorder the terms.

$$-adye^{-bx}+ace^{-bx}=\frac{\mathrm{d}(y)}{\mathrm{d}x}$$

Combine all terms containing a.

$$\left(-dye^{-bx}+ce^{-bx}\right)a=\frac{\mathrm{d}(y)}{\mathrm{d}x}$$

The equation is in standard form.

$$\frac{c-dy}{e^{bx}}a=0$$

Divide 0 by -dye^{-bx}+ce^{-bx}.

$$a=0$$

Solve for b

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

$$a=0\text{ or }\left(y=0\text{ and }c=0\right)\text{ or }\left(d=\frac{c}{y}\text{ and }y\neq 0\right)$$

Steps Using Definition of Logarithm

Use the distributive property to multiply ae^{\left(-b\right)x} by c-dy.

$$\frac{\mathrm{d}(y)}{\mathrm{d}x}=ae^{\left(-b\right)x}c-ae^{\left(-b\right)x}dy$$

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

$$ae^{\left(-b\right)x}c-ae^{\left(-b\right)x}dy=\frac{\mathrm{d}(y)}{\mathrm{d}x}$$

Reorder the terms.

$$-adye^{-bx}+ace^{-bx}=\frac{\mathrm{d}(y)}{\mathrm{d}x}$$

Combine all terms containing b.

$$\left(-ady+ac\right)e^{-bx}=\frac{\mathrm{d}(y)}{\mathrm{d}x}$$

Use the rules of exponents and logarithms to solve the equation.

$$\left(ac-ady\right)e^{\left(-x\right)b}=0$$

Divide both sides by -ady+ac.

$$e^{\left(-x\right)b}=0$$

Take the logarithm of both sides of the equation.

$$\log(e^{\left(-x\right)b})=\log(0)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$\left(-x\right)b\log(e)=\log(0)$$

Divide both sides by \log(e).

$$\left(-x\right)b=\frac{\log(0)}{\log(e)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$\left(-x\right)b=\log_{e}\left(0\right)$$

Divide both sides by -x.

$$b=\text{Indeterminate}$$