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🗝️ 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, \(y\text{.}\)
