Answer to Question #231177 in Differential Equations for Randal Rodriguez

Question

verify that each given function is a solution of the

differential equation.

8.y′′′′\:+\:4y′′′\:+\:3y\:=\:t;\:y\left(t\right)\:=\frac{t}{3}

Answer 1

$$y''''+4y'''+3y=t, y_1(t)=\dfrac{t}{3}$$

$$y=\dfrac{t}{3}=>y'=\dfrac{1}{3}, y''=y'''=y''''=0$$

Substitute

$$0+4(0)+3(\dfrac{t}{3})=t$$

$$t=t, True\ for\ t\in(-\infin, \infin)$$

The function $y_1(t)=\dfrac{t}{3}$ is a solution of the given differential equation.

$$y''''+4y'''+3y=t, y_2(t)=e^{-t}+\dfrac{t}{3}$$

$$y=e^{-t}+\dfrac{t}{3}=>y'=-e^{-t}+\dfrac{1}{3}, y''=e^{-t},$$

$$y'''=-e^{-t},y''''=e^{-t}$$

Substitute

$$e^{-t}+4(-e^{-t})+3(e^{-t}+\dfrac{t}{3})=t$$

$$t=t, True\ for\ t\in(-\infin, \infin)$$

The function $y_2(t)=e^{-t}+\dfrac{t}{3}$ is a solution of the given differential equation.