Chapter 7.1, Problem 79E

a.

To determine

To find: The general solution of the differential equation.

Answer to Problem 79E

The general solution of the differential equation

  $y=\frac{x^2}{2}+C$ .

Explanation of Solution

Given information: the differential equation

  $y^{\text{'}}=x$ .

Calculation:

the differential equation

  $y^{\text{'}}=x$ .

Replace $y^{\text{'}}\text{ with }\frac{dy}{dx}$

  $\begin{array}{l}{y}^{\text{'}}=x\\ \frac{dy}{dx}=x\\ {\displaystyle \int dy}={\displaystyle \int xdx}\\ y=\frac{x^2}{2}+C\end{array}$

b.

To determine

To find: The general solution of the differential equation.

Answer to Problem 79E

The general solution of the differential equation

  $y=-\frac{x^2}{2}+C$ .

Explanation of Solution

Given information: the differential equation

  $y^{\text{'}}=y$ .

Calculation:

the differential equation

  $y^{\text{'}}=y$

Replace $y^{\text{'}}\text{ with }\frac{dy}{dx}$

  $\begin{array}{l}{y}^{\text{'}}=-x\\ \frac{dy}{dx}=-x\\ {\displaystyle \int dy}=-{\displaystyle \int xdx}\\ y=-\frac{x^2}{2}+C\end{array}$

c.

To determine

To find: The general solution of the differential equation.

Answer to Problem 79E

The general solution of the differential equation

  $y=C{e}^x$ .

Explanation of Solution

Given information: the differential equation

  $y^{\text{'}}=y$ .

Calculation:

the differential equation

  $y^{\text{'}}=y$ .

Replace $y^{\text{'}}\text{ with }\frac{dy}{dx}$

  $\begin{array}{l}{y}^{\text{'}}=y\\ \frac{dy}{dx}=y\\ {\displaystyle \int \frac{dy}{y}}={\displaystyle \int dx}\\ \ln \left|y\right|=x+C\\ y=e^{x+C}\\ y=C{e}^x\end{array}$

d.

To determine

To find: The general solution of the differential equation.

Answer to Problem 79E

The general solution of the differential equation

  $y=C{e}^x$ .

Explanation of Solution

Given information: the differential equation

  $y^{\text{'}}=-y$ .

Calculation:

the differential equation

  $y^{\text{'}}=-y$

Replace $y^{\text{'}}\text{ with }\frac{dy}{dx}$

  $\begin{array}{l}{y}^{\text{'}}=-y\\ \frac{dy}{dx}=-y\\ -{\displaystyle \int \frac{dy}{y}}={\displaystyle \int dx}\\ -\ln \left|y\right|=x+C\\ y=e^{-x+C}\\ y=C{e}^{-x}\end{array}$

e.

To determine

To find: The general solution of the differential equation.

Answer to Problem 79E

The general solution of the differential equation

  $y=C{e}^x$ .

Explanation of Solution

Given information: the differential equation

  $y^{\text{'}}=xy$ .

Calculation:

the differential equation

  $y^{\text{'}}=xy$

Replace $y^{\text{'}}\text{ with }\frac{dy}{dx}$

  $\begin{array}{l}{y}^{\text{'}}=xy\\ \frac{dy}{dx}=xy\\ {\displaystyle \int \frac{dy}{y}}={\displaystyle \int xdx}\\ \ln \left|y\right|=\frac{x^2}{2}+C\\ y=e^{\frac{x^2}{2}+C}\\ y=C{e}^{\frac{x^2}{2}}\end{array}$