First-Order ODEs: Forms and Theorems

Section 15.2 First-Order ODEs: Forms and Theorems

First-order ODEs (Section 15.1) are among the simplest. For fairly general smoothness conditions, a solution is not only guaranteed to exist, but is also guaranteed to be unique. The technical theorem is stated below, but you may want to regard this as a Don’t Worry about It theorem. It tells you that you can just go ahead and try to solve it by one of the analytical methods listed in the next couple of sections. If you find a solution, then it exists and is unique. Usually, any limitations on WHERE it exists will be obvious from the physical situation.

There are a number of different methods for solving these equations. Which one you may want to use depends on the form of the equation. Listed here are some common forms of first-order ODEs:

Definition 15.5. Forms of First-Order ODEs.

Standard Form: You can ALWAYS write a first-order ODE in standard form:

\begin{equation} \frac{dy}{dx}=f(x,y).\tag{15.2.1} \end{equation}

In this form, you are isolating the derivative term on one side of the equation, which may involve some really messy inverse functions on the other side of the equation. Because it can be messy, this form is used primarily in the statement of theorems about existence and uniqueness of solutions.

Separable Form: If you can write

\begin{equation} f(x,y)=h(x)g(y),\tag{15.2.2} \end{equation}

then Eqn(15.2.1) is called separable. It is NOT always possible to write the equation in this form, but when you can, the solution is straightforward. See Section 15.3 for a discussion of the solution method.

Differential Form: If you write the standard form equation as

\begin{equation} f(x,y)=-\frac{M(x,y)}{N(x,y)},\tag{15.2.3} \end{equation}

then with some straightforward algebraic rearrangement, Eqn(15.2.1) becomes the differential form

\begin{equation} M(x,y)\, dx + N(x,y)\, dy =0\tag{15.2.4} \end{equation}

There are many ways to divide \(f(x,y)\) into \(M(x,y)\) and \(N(x,y)\text{.}\) Choose a way that is helpful for the problem at hand. In particular, you can ALWAYS choose \(M(x,y)\) and \(N(x,y)\) so that (15.2.4) is an exact differential and the equation can be solved by simple integration. See Section 15.4 for a discussion of this solution method.

Theorem 15.6. First-Order ODEs: Uniqueness Theorem.

If \(f\) and \(\frac{\partial f}{\partial y}\) are continuous in a rectangle \(\vert x-x_0\vert \le a\text{,}\) \(\vert y-y_0\vert\le b\text{,}\) then there exists an interval about \(x_0\) in which the initial value problem \(y^{\prime}=f(x,y)\text{,}\) \(y(x_0)=y_0\) has a unique solution.

To Remember.

Under physically reasonable circumstances, the solution of a first-order ODE exists and is unique. If you can find a solution by any convenient method, you can be confident that it is THE solution.

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