Question

In Problems 41 and 42 verify that the indicated pair of functions is a solution of the given system of differential equations on the interval .

$$\begin{gathered} \frac{dy}{dx}=x+3y \\ \frac{dy}{dt}=5x+3y; \\ x=e^{-2t}+3e^{6t}, \\ y=-e^{-2t}+5e^{6t} \end{gathered}$$

Short answer

The derivatives of the equations and substitute in the differential equations to verify that they are the solutions.

Step-by-step solution

Definition of differential equation.

A differential equation is defined as the derivative function of one or more unknown functions (dependable variables) with respect to one or more undependable variables.

First derivative of x and y.

Given$x=e^{-2t}+3e^{6t}$

Finding the first derivation of$x$,

$\frac{dx}{dt}=-2e^{-2t}+18e^{6t}$

$y=-e^{-2t}+5e^{6t}$

Finding the first derivative of$y$,

$\frac{dy}{dt}=2e^{-2t}+30e^{6t}$

Substituting to the first differential equation and simplify.

Given that,$\frac{dy}{dx}=x+3y$

$-2e^{-2t}+18e^{6t}=[e^{-2t}+3e^{6t}]+3[-e^{-2t}+5e^{6t}]$

Simplify,

$-2e^{-2t}+18e^{6t}=-2e^{-2t}+18e^{6t}$

The solution is verified.

Since the solution satisfies the system of differential equations, the solution is verified.

Substituting to the second differential equation and simplify.

Given that,

$$\begin{gathered} \frac{dy}{dt}=5x+3y \\ 2e^{-2t}+30e^{6t}=5[e^{-2t}+3e^{6t}]+3[-e^{-2t}+5e^{6t}] \end{gathered}$$

Simplify,

$$\begin{gathered} 2e^{-2t}+30e^{6t}=5e^{-2t}+15e^{6t}-3e^{-2t}+15e^{6t} \\ 2e^{-2t}+30e^{6t}=2e^{-2t}+30e^{6t} \end{gathered}$$

Therefore, The solution is verified. Since the solution satisfies the system of differential equations, the solution is ver