Review / Part I — Explicit & Qualitative Methods / Ch 1

What is an ODE?

Ch 1 — Part I — Explicit & Qualitative Methods

A differential equation is an equation relating an unknown function to its derivatives. A solution is a function that satisfies the equation at every point in its domain.

We study ordinary differential equations (ODEs) — equations with a single independent variable (typically \(t\) or \(x\)). Partial differential equations involve several independent variables and are covered in Part IV.

A guiding example

The equation \(y'(t) = r\,y(t)\) says that the rate of change of \(y\) is proportional to \(y\) itself — exponential growth when \(r>0\), decay when \(r<0\). The general solution is $$y(t) = C e^{rt}, \qquad C \in \mathbb{R}.$$ Every such ODE of this form shows up in population growth, radioactive decay, Newton's law of cooling, and continuously compounded interest.

Checking a candidate solution

To verify that a function \(y(t)\) is a solution of an ODE, plug it (and its derivatives) into the equation and check that both sides agree for every \(t\) in the relevant interval. Solving and checking are different skills — always end by checking.

Where they come from

ODEs arise whenever a system's change is determined by its current state: mechanics (Newton's second law), electrical circuits, chemical kinetics, epidemiology (SIR), finance, biology (logistic growth). The model specifies the right-hand side; our job is to analyze the resulting equation.

Practice Problems

Problem 1easy
Verify that \(y(t) = 5e^{2t}\) is a solution of \(y'(t) = 2y(t)\).
Hint
Differentiate \(y(t)\) directly, then compare with \(2y(t)\).
Solution

We have \(y'(t) = 5 \cdot 2 e^{2t} = 10 e^{2t}\). Also \(2y(t) = 2(5e^{2t}) = 10e^{2t}\). Both agree for all \(t\).

Answer: Yes, it satisfies the ODE.

Problem 2easy
Is \(y(x) = x^2\) a solution of \(xy'(x) - 2y(x) = 0\) on \(\mathbb{R}\)?
Hint
Compute \(y'\) and substitute into \(xy'-2y\).
Solution

\(y'(x) = 2x\), so \(xy' = 2x^2\) and \(2y = 2x^2\). Thus \(xy' - 2y = 0\) for every \(x\).

Answer: Yes.

Problem 3medium
For what constants \(r\) does \(y(t) = e^{rt}\) satisfy \(y'' - 3y' + 2y = 0\)?
Hint
Substitute and factor the resulting polynomial in \(r\).
Solution

Computing \(y'' - 3y' + 2y = (r^2 - 3r + 2)e^{rt}\). Since \(e^{rt} \ne 0\), we need \(r^2 - 3r + 2 = 0\), i.e. \((r-1)(r-2)=0\).

Answer: \(r = 1\) or \(r = 2\).

Problem 4easy
Classify the equation \(y''' + \sin(t) y' + y = e^t\) as ODE or PDE, and identify whether the unknown is a single-variable function.
Hint
Count independent variables.
Solution

Only one independent variable \(t\) appears, so it is an ODE. The unknown \(y(t)\) depends only on \(t\).

Answer: Third-order ODE in one variable.