Question

In Exercises \(1-10,\) find the general solution to the exact differential equation. $$\frac{d y}{d x}=\sec x \tan x-e^{x}$$

Short answer

The general solution to the differential equation is \(y = \sec(x)^{2}/2 - e^{x} + C\).

Step-by-step solution

Identify the ODE type

The given differential equation is in the form of \(\frac{d y}{d x} = f(x)\) where \(f(x) = \sec x \tan x - e^{x}\). This is a first order separable differential equation.

Rearrange the equation

Rearrange the equation, such that terms involving the same variable are on the same side of the equation. We can write the ODE as \(dy = (\sec(x)\tan(x) - e^{x})dx\)

Integrate both sides

The next step is to integrate both sides of the equation. The integral of the left side with respect to \(y\) is just \(y\). The integral of the right side with respect to \(x\) gives us \(y = \int (\sec(x)\tan(x) - e^{x})dx\).

Calculate the Integral

Calculate the integral: \(y = \int \sec(x)\tan(x)dx - \int e^{x}dx = \int \sec(x) d(\sec(x)) - \int e^{x} dx = \sec(x)^{2}/2 - e^{x} + C\), where \(C\) is the constant of integration.