Question

Answer Problems 1-12without referring back to the text. Fill in the blanks or answer true or false.

If ${\mathit{y}}^{\cancel{\mathbf{c}}}\mathbf{=}{\mathit{e}}^{\mathbf{x}}\mathit{y}$, then y =

Short answer

The solution is y = c₁e^ex.

Step-by-step solution

Definition

A first-order differential equation of the form $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathit{g}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathit{h}\mathbf{\left(}\mathbf{y}\mathbf{\right)}$is said to be separable or to have separable variables.

Separating variables

The given equation is separable.

$$\begin{gathered} \frac{dy}{dx}=e^{x}y \\ \frac{dy}{y}=e^{x}dx \\ \int \frac{dy}{y}=\int e^{x}dx \\ Iny=e^x+c \end{gathered}$$

Solve for y

In order to solve fortake antilog both sides.

$$\begin{gathered} y=e^{e^x+c} \\ y=c_1{e}^{e^x} \end{gathered}$$

So, if y' = e^xy the value of y = c₁e^ex.