Question

In Exercises \(1-10,\) find the general solution to the exact differential equation. $$\frac{d u}{d x}=\left(\sec ^{2} x^{5}\right)\left(5 x^{4}\right)$$

Short answer

The general solution to the exact differential equation is \(u = \tan(x^5) + C\).

Step-by-step solution

Rewrite the Equation

We can express the derivative as an integral to solve the problem. In this case, the differential equation \(\frac{d u}{d x}=\sec^{2}\(x^5)\(5x^4)\) can be rewritten in terms of u. Hence, we have \(du = \sec^{2}\(x^5)\(5x^4)dx\).

Compute the Integral

Use the integral rules to compute the integral of the right-hand side of the equation. Here, the integral is simply taking the antiderivative of the function with respect to x, which can be calculated as \(u = \int \sec^{2}\(x^5)\(5x^4)dx\).

Calculate the Antiderivative

Calculate the antiderivative. The function is a known composite function. The derivative of \(\tan(x)\) is \(\sec^2(x)\). Here's what we have: \(x^5\) replacing \(x\) in \(\tan(x)\). Thus, the antiderivative is \(\tan(x^5)\) and the integral of \(\frac {d}{dx}\) of \(x^4\), which is \(x^5\). So, \(u = \tan(x^5) + C\), where C is the constant of integration.