DIFFERENTIAL EQUATIONS · ONE INDEPENDENT VARIABLE
Ordinary Differential Equations
An ODE turns a law of change into a function. The key skill is not memorizing one formula, but recognizing the structure of the equation and matching it to a valid method.
1. ODEs and initial-value problems
An ordinary differential equation contains derivatives with respect to a single independent variable. A general first-order ODE may be written
If an initial condition \(y(x_0)=y_0\) is supplied, we have an initial-value problem. The differential equation gives a family of possible solution curves; the initial condition selects one curve.
2. Classify before you solve
3. Separable first-order equations
If the equation can be written as \(y'=g(x)h(y)\), then—where division by \(h(y)\) is legitimate—we may separate variables:
Integrating both sides gives an implicit or explicit family of solutions.
Worked example: exponential growth
Solve \(y'=ky\), \(y(0)=y_0\). For nonzero \(y\),
The initial condition gives \(C=y_0\), so
4. Linear first-order equations
A first-order linear equation has the standard form
The integrating factor
turns the left side into a product derivative:
Worked example
Solve \(y'+2y=4\). Here \(\mu=e^{2x}\), so
Integrating gives \(e^{2x}y=2e^{2x}+C\), hence
The constant solution \(y=2\) is the equilibrium toward which all solutions converge as \(x\to\infty\).
5. Autonomous equations and stability
For \(y'=f(y)\), an equilibrium \(y_*\) satisfies \(f(y_*)=0\). The sign of \(f(y)\) tells us whether nearby solutions move upward or downward.
For the logistic equation \(y'=ry(1-y/K)\), the equilibria are \(y=0\) and \(y=K\). For \(r>0\), \(0\) is unstable and \(K\) is stable.
6. Second-order linear equations
A constant-coefficient homogeneous equation has the form
Trying \(y=e^{rx}\) produces the characteristic equation
| Roots | General solution |
|---|---|
| Distinct real \(r_1,r_2\) | \(y=C_1e^{r_1x}+C_2e^{r_2x}\) |
| Repeated real \(r\) | \(y=(C_1+C_2x)e^{rx}\) |
| Complex \(\alpha\pm i\beta\) | \(y=e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)\) |
Example: harmonic oscillator
For \(y''+4y=0\), the roots are \(r=\pm2i\), hence
7. Interactive slope-field explorer
Explore three first-order ODEs. The short line segments show the local slope \(y'=f(x,y)\), while the highlighted curve is a numerical solution through \((0,y_0)\).
8. Method selection checklist
Try to isolate all \(y\)-dependence with \(dy\) and all \(x\)-dependence with \(dx\).
Put it into \(y'+p(x)y=q(x)\) and use an integrating factor.
Find equilibria and use a phase-line analysis even when an explicit formula is difficult.
Use the characteristic polynomial and classify its roots.
9. Practice
- Solve \(y'=3xy\) with \(y(0)=2\).
- Solve \(y'-y=e^{2x}\).
- Find the equilibria of \(y'=y(2-y)\) and classify their stability.
- Solve \(y''-5y'+6y=0\).
Solutions
1) \(y=2e^{3x^2/2}\). 2) \(y=e^{2x}+Ce^x\). 3) Equilibria: \(0\) unstable and \(2\) stable. 4) Characteristic roots \(2,3\), so \(y=C_1e^{2x}+C_2e^{3x}\).
10. Continue the learning path
Once scalar ODEs are familiar, the natural next step is to study coupled systems, where linear algebra and eigenvalues control the geometry and stability of solutions.