Solve i=I(e^ct-1)

$$i = I ( e ^ { ct } - 1 )$$

Solve for I

$$I=\frac{-i}{1-e^{ct}}$$

$$t\neq 0\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }c=\frac{2\pi n_{1}i}{t}$$

Steps for Solving Linear Equation

Use the distributive property to multiply I by e^{ct}-1.

$$i=Ie^{ct}-I$$

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

$$Ie^{ct}-I=i$$

Combine all terms containing I.

$$\left(e^{ct}-1\right)I=i$$

Divide both sides by e^{ct}-1.

$$\frac{\left(e^{ct}-1\right)I}{e^{ct}-1}=\frac{i}{e^{ct}-1}$$

Dividing by e^{ct}-1 undoes the multiplication by e^{ct}-1.

$$I=\frac{i}{e^{ct}-1}$$

Solve for c

$$c=\frac{2\pi n_{1}i}{t}+\frac{\ln(\frac{I+i}{I})}{t}$$

$$n_{1}\in \mathrm{Z}$$

$$t\neq 0\text{ and }I\neq 0\text{ and }I\neq -i$$