Solve y=ae^5x+5e^-5x

$$y = a e ^ { 5 x } + 5 e ^ { - 5 x }$$

Solve for a

$$a=\frac{ye^{5x}-5}{e^{10x}}$$

Steps for Solving Linear Equation

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

$$ae^{5x}+5e^{-5x}=y$$

Subtract 5e^{-5x} from both sides.

$$ae^{5x}=y-5e^{-5x}$$

The equation is in standard form.

$$e^{5x}a=-\frac{5}{e^{5x}}+y$$

Divide both sides by e^{5x}.

$$\frac{e^{5x}a}{e^{5x}}=\frac{-\frac{5}{e^{5x}}+y}{e^{5x}}$$

Dividing by e^{5x} undoes the multiplication by e^{5x}.

$$a=\frac{-\frac{5}{e^{5x}}+y}{e^{5x}}$$

Divide y-\frac{5}{e^{5x}} by e^{5x}.

$$a=\frac{ye^{5x}-5}{e^{10x}}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{5\ln(-\sqrt[5]{\frac{-\sqrt{y^{2}-20a}-y}{a}})-\ln(2)}{5}\text{, }&a\leq \frac{y^{2}}{20}\text{ and }a>0\text{ and }y\geq 2\sqrt{5a}\\x=\frac{5\ln(-\sqrt[5]{\frac{\sqrt{y^{2}-20a}-y}{a}})-\ln(2)}{5}\text{, }&\left(y=2\sqrt{5a}\text{ and }a>0\right)\text{ or }\left(y\geq 2\sqrt{5a}\text{ and }a\leq \frac{y^{2}}{20}\text{ and }a>0\right)\text{ or }a<0\\x=\frac{-\ln(y)+\ln(5)}{5}\text{, }&y>0\text{ and }a=0\end{matrix}\right.$$