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Chapter 4 Direct Integration
📝: 🎧 Listen.
Solving a differential equation might sound like a new challenge, but you’ve already done it in calculus! Whenever you find an antiderivative, you’re solving a differential equation.
In this section, we’ll reframe antiderivatives as solutions to differential equations and use this perspective to introduce our first method: direct integration. By the end, you’ll be able to find general and particular solutions to simple differential equations using integration.
🗝️ Key Takeaways...
The general solution to the differential equation
\begin{equation*}
\frac{dy}{dx} = f(x)
\end{equation*}
is given by
\begin{equation*}
y(x) = F(x) + c
\end{equation*}
where \(F(x)\) is any antiderivative of \(f(x)\) and \(c\) is any constant.
If the differential equation can be expressed as
\begin{equation*}
\frac{d}{dx}[\ ... ] = f(x)
\end{equation*}
then the general solution can be found by direct integration . For example, integrating both sides of the equations
\begin{equation*}
\frac{dy}{dx} = \cos x \quad\text{or}\quad \frac{d}{dx}\left[x^2\cos(y)\right] = x + e^x \text{.}
\end{equation*}
removes the derivative, allowing you to isolate the unknown, \(y\text{.}\)
