First-Order ODEs¶
Job: Solve or reduce scalar first-order equations; recognize exactness, integrating factors, and separation; track domains of existence.
Core¶
Normal form \(y'=f(x,y)\) or \(M(x,y)\,dx+N(x,y)\,dy=0\).
| Class | Structure | Method |
|---|---|---|
| Separable | \(y'=g(x)h(y)\) | \(\int dy/h=\int g\,dx\) (zeros of \(h\) are equilibria) |
| Linear | \(y'+P(x)y=Q(x)\) | \(\mu=\exp\int P\); \((\mu y)'=\mu Q\) |
| Exact | \(M_y=N_x\) | \(F\) with \(dF=M\,dx+N\,dy\); level sets \(F=C\) |
| Homogeneous | \(f(tx,ty)=f(x,y)\) | \(v=y/x\) reduces to separable |
| Bernoulli | \(y'+P y=Q y^n\) | \(z=y^{1-n}\) linearizes (\(n\neq0,1\)) |
| Riccati | \(y'=P+Qy+Ry^2\) | one particular solution \(\Rightarrow\) linear via \(y=y_p+1/z\) |
Integrating factor. If \((M_y-N_x)/N\) depends only on \(x\), then \(\mu(x)=\exp\int\frac{M_y-N_x}{N}\,dx\) makes \(\mu M\,dx+\mu N\,dy\) exact (symmetric criterion with roles reversed).
Clairaut: \(y=xy'+f(y')\) — general solution lines; singular solution by envelope (discriminant of the family).
Existence domain: after separation, the implicit solution may leave the region where \(h(y)\neq0\) or \(f\) is Lipschitz; check back against the DE.
Comparison / Gronwall. If \(\dot u\le a(t)u+b(t)\), then $$ u(t)\le u(t_0)e^{\int a}+\int_{t_0}^t b(s)e^{\int_s^t a}\,ds. $$ Used for a priori bounds and continuous dependence.
See it¶
Linear IVP \(y'+2xy=x\), \(y(0)=1\): \(\mu=e^{x^2}\), so $$ (e^{x^2}y)'=x e^{x^2}\Rightarrow y=e^{-x^2}\Bigl(\tfrac12 e^{x^2}+C\Bigr)=\tfrac12+C e^{-x^2}, $$ hence \(y=\frac12+\frac12 e^{-x^2}\). Global, unique by Picard (linear growth).
Check¶
When does \(M\,dx+N\,dy=0\) admit a \(\mu(x)\) integrating factor?
Iff \((M_y-N_x)/N\) is a function of \(x\) alone. Then \(\mu=\exp\int(M_y-N_x)/N\,dx\). If \((N_x-M_y)/M\) depends only on \(y\), use \(\mu(y)\).
Counterexample to uniqueness for continuous \(f\).
\(y'=3y^{2/3}\), \(y(0)=0\): \(y\equiv0\) and \(y=t^3\) (and pieced solutions through the origin). Hölder \(2/3<1\) \(\Rightarrow\) not Lipschitz.
Pitfalls¶
- Dividing by \(h(y)\) loses constant solutions \(y\equiv y_*\) with \(h(y_*)=0\).
- Exactness is local; monodromy of \(F\) on non-simply-connected domains can obstruct global first integrals.
- Homogeneous substitution \(v=y/x\) fails on \(x=0\); treat as a separate chart or use \(x=vy\).
Refs¶
Coddington; Ince, Ordinary Differential Equations (classical special forms); Arnold (geometric viewpoint).