Chapter 4.1, Problem 1P

Program Plan Intro

Program Description:Purpose of problem is to transform the differential equation $x^{″}+3x^{\prime }+7x=t^2$ into an equivalent system of first order differential equations.

Explanation of Solution

Given information:

The differential equation is $x^{″}+3x^{\prime }+7x=t^2$ .

Explanation:

Thedifferential equation is written as,

  $x^{″}=t^2-3x^{\prime }-7x$   .......... (1)

Consider the expression given below.

  $x=x_1$

Differentiate the above expression.

  $x_2={x^{\prime }}_1$

Substitute x for $x_1$ in above equation,

  $x_2=x^{\prime }$

Differentiate the above equation.

  ${x^{\prime }}_2={x^{″}}_1$

Substitute x for $x_1$ in above equation,

  ${x^{\prime }}_2=x^{″}$

Substitute $x_2$ for $x^{\prime }$ , $x_1$ for x and ${x^{\prime }}_2$ for $x^{″}$ in equation (1).

  ${x^{\prime }}_2=t^2-3x_2-7x_1$

Conclusion:

Thus, the transformation of the differential equation $x^{″}+3x^{\prime }+7x=t^2$ into an equivalent system of first order differential equations is ${x^{\prime }}_1=x_2\text{ and }{x^{\prime }}_2=-7x_1-3x_2+t^2$ .