Chapter 7.1, Problem 1E

To determine

To find the general solution to the exact differential equation.

Answer to Problem 1E

  $\text{The general solution is, }y=x^5-\tan x+c$

Explanation of Solution

Given information:

  $\text{The given differential equation is, }\frac{dy}{dx}=5x^4-{\sec }^{2}x.$

Calculation: The general solution is composed of the antiderivative of the RHS of the exact differential equation. This involves using the power law in reverse.

Look for the expressions that look like the result of the chain rule.

  $\begin{array}{l}\frac{dy}{dx}=5x^4-{\sec }^{2}x\\ dy=\left(5x^4-{\sec }^{2}x\right)dx\\ {\displaystyle \int dy=}{\displaystyle \int \left(5x^4-{\sec }^{2}x\right)dx}\left(\text{taking integral both side}\right)\\ y=x^5-\tan x+c\\ \text{Hence, the general solution is, }y=x^5-\tan x+c\end{array}$