Chapter 7.1, Problem 78E

a.

To determine

To find: The specific solution of the second order differential equation.

Answer to Problem 78E

The specific solution of the second order differential equation

  $y=2x^4-5x^2+5x+3$ .

Explanation of Solution

Given information: second order differential equation

  $\frac{d^{2}y}{d{x}^2}=24x^2-10.\text{When }x=1,\frac{dy}{dx}=3\text{ and }y=5$

Calculation: second order differential equation

  $\frac{d^{2}y}{d{x}^2}=24x^2-10.\text{When }x=1,\frac{dy}{dx}=3\text{ and }y=5$

Anti-differentiate to find $\frac{dy}{dx}$

  $\begin{array}{l}\frac{d^{2}y}{d{x}^2}=24x^2-10\\ \frac{dy}{dx}=8x^3-10x+C_1\\ \text{Given }\\ x=1,\frac{dy}{dx}=3\\ 3=8-10+C_1\\ C_1=5\\ \frac{dy}{dx}=8x^3-10x+5\\ {\displaystyle \int dy}={\displaystyle \int \left(8x^3-10x+5\right)}dx\\ y=2x^4-5x^2+5x+C_2\\ \text{Given }\\ x=1,y=5\\ 5=2+C_2\\ C_2=3\\ y=2x^4-5x^2+5x+3\end{array}$

b.

To determine

To find: The specific solution of the second order differential equation.

Answer to Problem 78E

The specific solution of the second order differential equation

  $y=\cos x-\sin x+3x+1$ .

Explanation of Solution

Given information: second order differential equation

  $\frac{d^{2}y}{d{x}^2}=\cos x-\sin x.\text{When }x=0,\frac{dy}{dx}=2\text{ and }y=2$

Calculation: second order differential equation

  $\frac{d^{2}y}{d{x}^2}=\cos x-\sin x.\text{When }x=0,\frac{dy}{dx}=2\text{ and }y=2$

Anti-differentiate to find $\frac{dy}{dx}$

  $\begin{array}{l}\frac{d^{2}y}{d{x}^2}=\cos x-\sin x\\ \frac{dy}{dx}=\sin x+\cos x+C_1\\ \text{Given }\\ x=0,\frac{dy}{dx}=2\\ 2=1+C_1\\ C_1=1\\ \frac{dy}{dx}=\sin x+\cos x+1\\ {\displaystyle \int dy}={\displaystyle \int \left(\sin x+\cos x+1\right)}dx\\ y=-\cos x+\sin x+x+C_2\\ \text{Given }\\ x=0,y=0\\ 0=-1+C_2\\ C_2=1\\ y=-\cos x+\sin x+x+1\end{array}$

c.

To determine

To find: The specific solution of the second order differential equation.

Answer to Problem 78E

The specific solution of the second order differential equation

  $y=e^x-\frac{x^3}{6}-x+1$ .

Explanation of Solution

Given information: second order differential equation

  $\frac{d^{2}y}{d{x}^2}=e^x-x.\text{When }x=0,\frac{dy}{dx}=0\text{ and }y=1$

Calculation: second order differential equation

  $\frac{d^{2}y}{d{x}^2}=e^x-x.\text{When }x=0,\frac{dy}{dx}=0\text{ and }y=1$

Anti-differentiate to find $\frac{dy}{dx}$

  $\begin{array}{l}\frac{d^{2}y}{d{x}^2}=e^x-x\\ \frac{dy}{dx}=e^x-\frac{x^2}{2}+C_1\\ \text{Given }\\ x=0,\frac{dy}{dx}=0\\ 0=1+C_1\\ C_1=-1\\ \frac{dy}{dx}=e^x-\frac{x^2}{2}-1\\ {\displaystyle \int dy}={\displaystyle \int \left(e^x-\frac{x^2}{2}-1\right)}dx\\ y=e^x-\frac{x^3}{6}-x+C_2\\ \text{Given }\\ x=0,y=1\\ 1=1+C_2\\ C_2=0\\ y=e^x-\frac{x^3}{6}-x\end{array}$