Ordinary Differential Equations

In Calculus 3, we are often interested in level curves of the form \(F(t,y)=C\text{,}\) where \(C\) is an arbitrary constant. If we differentiate with respect to \(t\text{,}\) we get \(\dfrac{\mathrm{d}}{\mathrm{d}t} F(t,y) = \dfrac{\mathrm{d}}{\mathrm{d}t} C\) which simplifies (with "partial derivative" notation) to \(\dfrac{\partial F}{\partial t} + \dfrac{\partial F}{\partial y} \dfrac{\mathrm{d}y}{\mathrm{d}t}=0\text{.}\) Solving this for \(\dfrac{\mathrm{d}y}{\mathrm{d}t}\) then yields the differential equation

Some first-order nonlinear, nonseparable ODEs can be solved by hand. To do so, we interpret (1.12.1) in differential form as

We use (1.12.2) to look for (implicit) solutions of the form \(F(t,y)=C\text{.}\)

Definition 1.12.1. Exact equations.

The differential form (1.12.2) is said to be exact in a region \(R\) in the plane if there exists a function \(F\) such that \(\dfrac{\partial F}{\partial t}=M\) and \(\dfrac{\partial F}{\partial y}=N\) for all \((t,y) \in R\text{.}\)

🔗

🔗

Theorem 1.12.2. Test for exactness.

Suppose \(M\) and \(N\) have continuous first-order partial derivatives in a region \(R\) in the plane. Then (1.12.2) is exact on \(R\) if and only if \(\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial t}\) for all \((t,y) \in R\text{.}\)

🔗

🔗

Algorithm 1.12.3.

Let (1.12.2) be exact. To solve it, we do the following.

1. Integrate \(M\) with respect to \(t\) to get \(F(t,y)=\displaystyle\int M(t,y) \mathrm{d}t + g(y)\) for some unknown function \(g\text{.}\)
   🔗
2. Integrate \(N\) with respect to \(y\) to get \(F(t,y)=\displaystyle\int N(t,y) \mathrm{d}y + h(t)\text{,}\) for some unknown function \(h\text{.}\)
   🔗
3. Compare like and unique terms to find \(F(t,y)\text{.}\)
   🔗
4. Write an implicit solution of the form \(F(t,y)=C\text{.}\)
   🔗

🔗

🔗

Example 1.12.4.

Consider the differential form \(\left(e^t \sin(y)+2t\right)\mathrm{d}t + \left(e^t \cos(y)+3 \right)\mathrm{d}y=0\text{.}\)

1. Find the order of the differential equation. Is it (non)linear, (non)autonomous, and (non)separable?
   🔗
2. What method(s) can you use to solve this differential equation?
   🔗
3. Find an implicit solution.
   🔗

🔗

🔗

Example 1.12.5.

Consider the differential form \(\left(y^2+2ty\right)\mathrm{d}t - t^2 \mathrm{d}y=0\text{.}\) Show this equation is not exact.

🔗

🔗

Definition 1.12.6.

If the differential form (1.12.2) is not exact, but there is a function \(\mu\) so that

\begin{equation*} \mu(t,y)M(t,y)\mathrm{d}t + \mu(t,y)N(t,y)\mathrm{d}y=0, \end{equation*}

then the function \(\mu\) is called an integrating factor for (1.12.2). When using such an integrating factor, it is possible to gain or lose solutions.

🔗

🔗

Theorem 1.12.7.

Suppose that the differential form (1.12.2) is not exact.

🔗

If \(\dfrac{\frac{\partial M}{\partial y}-\frac{\partial N}{\partial t}}{N}\) is continuous and depends only on \(t\text{,}\) then we can take the integrating factor to be

\begin{equation*} \mu(t,y)\equiv \mu(t) = \exp \left( \displaystyle\int \dfrac{\frac{\partial M}{\partial y}-\frac{\partial N}{\partial t}}{N} \mathrm{d}t \right). \end{equation*}

🔗

If \(\dfrac{\frac{\partial N}{\partial t} - \frac{\partial M}{\partial y}}{M}\) is continuous and depends only on \(y\text{,}\) then we can take the integrating factor to be

\begin{equation*} \mu(t,y)\equiv \mu(y) = \exp \left( \dfrac{\frac{\partial N}{\partial t} - \frac{\partial M}{\partial y}}{M} \right). \end{equation*}

🔗

🔗

Example 1.12.8.

Consider the differential form \(\left(2t^2+y\right)\mathrm{d}t+\left(t^2y-t\right)\mathrm{d}y=0\text{.}\)

1. Find the order of the differential equation. Is it (non)linear, (non)autonomous, and (non)separable?
   🔗
2. What method(s) can you use to solve this differential equation?
   🔗
3. Find an implicit solution.
   🔗

🔗

🔗

Example 1.12.9.

Consider the differential form \(2ty \mathrm{d}t+\left(y^2-3t^2\right)\mathrm{d}y=0\text{.}\)

1. Find the order of the differential equation. Is it (non)linear, (non)autonomous, and (non)separable?
   🔗
2. What method(s) can you use to solve this differential equation?
   🔗
3. Find an implicit solution.
   🔗

🔗

🔗

Example 1.12.10.

Consider the differential form \(\left(3y+3y^3\right)\mathrm{d}t+\left(ty^2-t\right)\mathrm{d}y=0\text{.}\)

1. Show that this equation is not exact.
   🔗
2. Show that this equation becomes exact when both sides are multiplied by the integrating factor \(\mu(t,y)=\dfrac{t^2}{y^2}\text{.}\) Find an implicit solution.
   🔗

🔗

🔗