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Section 1.10 Separable differential equations
Recall from
Section 1.1 that first-order separable differential equations take the form
(1.1.2) . We solve such ODEs by the
separation of variables algorithm .
Algorithm 1.10.1 .
To solve
(1.1.2) , we "illegally" treat
\(\dfrac{\mathrm{d}y}{\mathrm{d}t}\) as a fraction and rewrite it as
\begin{equation*}
\dfrac{1}{h(y)} \mathrm{d}y = g(t) \mathrm{d}t
\end{equation*}
and apply an integral on the left to each side to get
\begin{equation*}
\displaystyle\int \dfrac{1}{h(y)} \mathrm{d}y = \displaystyle\int g(t) \mathrm{d}t.
\end{equation*}
This is technically an illegal mathematical step because the operation of integration includes its differential, but it nonetheless still works!
Example 1.10.2 .
Classify and solve
\(ay'+by=0\text{.}\)
Example 1.10.3 .
Continuing with
Example 1.9.7 , we ask what amount of the compound
\(X\) there is after
\(20\) minutes.
Example 1.10.4 .
Consider the differential equation \(4t\ln(y)\dfrac{\mathrm{d}y}{\mathrm{d}t} = \dfrac{\sqrt{t^2-4}}{y^3}\text{.}\)
Find the order of the differential equation. Is it (non)linear, (non)automous, and (non)separable?
What method(s) can you use to solve this differential equation?
Find an implicit solution to the differential equation.
Example 1.10.5 .
Consider the differential equation \(2y\dfrac{\mathrm{d}y}{\mathrm{d}t} = \dfrac{\sec(y^2)}{1+t^2}, \quad y(1)=0\text{.}\)
Find the order of the differential equation. Is it (non)linear, (non)automous, and (non)separable?
What method(s) can you use to solve this differential equation?
Solve the initial value problem.
