Chapter 6.1, Problem 1E

To determine

To find: the general solution to the given exact differential equation.

Answer to Problem 1E

  $y=x^5-\tan x+C$ .

Explanation of Solution

Given:

  $\frac{dy}{dx}=5x^4-{\sec }^{2}x$ .

Concept used:

The general solution of the exact differential equation of the form $\frac{dy}{dx}=f\left(x\right)$ is $y={\displaystyle \int f\left(x\right)dx}$ .

Calculation:

Consider the given differential equation,

  $\frac{dy}{dx}=5x^4-{\sec }^{2}x$

Now the general solution is,

  $\begin{array}{c}y={\displaystyle \int \left(5x^4-{\sec }^{2}x\right)dx}\\ y=5\cdot \frac{x^5}{5}-\tan x+C\\ y=x^5-\tan x+C\end{array}$