MTH2032 Chap.7 Characteristics for First-Order PDEs
Characteristics for First-Order PDEs
Week 7 solves first-order PDEs by following their characteristic curves. For a semilinear equation the characteristic curves solve dy/dx = B/A, and changing to coordinates that follow them turns the PDE into an ODE along each curve, the canonical form.
For a quasilinear equation the slope depends on u, so characteristics live in three dimensions and the general solution links two independent quantities that stay constant along them. Both routes are shown on original examples, each checked by substitution.
What this chapter covers
- 01
Standard form A u_x + B u_y = C
- 02
Characteristic ODE dy/dx = B/A
- 03
Choosing s and τ
- 04
Canonical form and inversion
- 05
Characteristic equations in (x, y, u)
- 06
Intersecting surfaces approach
Worked example · free
Growth along diagonal characteristics
- 1A = B = 1, so dy/dx = 1 and the characteristics are y − x = s.
- 1Choose τ = x, whose level lines cross every characteristic once.
- 1Along a characteristic u_x + u_y is the τ-derivative, so ū_τ = ū and ū = F(s)e^τ.
- 1Return to the original variables: u = F(y − x)e^x.
Key terms
- Transformation Method
- The change of variables to coordinates (s, τ) along which a semilinear first-order PDE becomes an ODE.
- First Integral
- A combination of x, y and u that stays constant along every characteristic of a quasilinear PDE.
- Characteristic Direction
- The vector (A, B), or (A, B, C) for a quasilinear equation, along which the solution is traced.
- Solution Surface
- The graph u = u(x, y) of a solution, tangent at every point to the characteristic direction.
- Characteristic ODE
- The ordinary differential equation that a first-order PDE becomes along each characteristic after the change of coordinates.
Characteristics for First-Order PDEs FAQ
How do I choose the second coordinate τ?
Any differentiable function works if its level curves are never parallel to the characteristics and each characteristic meets each level curve exactly once, so that the change of variables can be inverted. Taking τ = x or τ = y is usually enough; say in your answer why the choice is acceptable.
What is different about quasilinear equations?
Their coefficients depend on u, so the slope dy/dx = B/A cannot be computed before the solution is known. The geometric method treats characteristics as curves in (x, y, u) space solving dx/dτ = A, dy/dτ = B and du/dτ = C, finds two independent first integrals, and writes one as an arbitrary function of the other.
Why can two general solutions look different?
Arbitrary functions can absorb terms. For u_x + u_y = 1 the notes show that x + F(y − x) and y + G(y − x) describe the same set of solutions, since y − x is constant on each characteristic and G can absorb the difference. Check equivalence by substituting both into the PDE.
How do I check a general solution with an arbitrary function?
Differentiate the formula using the chain rule, leaving the arbitrary function as F and its derivative as F′, and substitute into the PDE. Every F′ term should cancel or combine into the right-hand side. If F′ survives, the characteristic coordinate inside F is wrong, which usually means the slope dy/dx was inverted.
Exam move
Learn the seven-step transformation algorithm as a checklist and use it on every semilinear problem until the steps are automatic: identify A, B and C, solve dy/dx = B/A, set s, choose and justify τ, transform, solve the ODE in τ, invert. Always finish with a substitution check, which takes two lines and catches most chain-rule errors. Then practise the quasilinear method in both forms.
In the parametric approach, solve the three characteristic equations, keep the three constants, and eliminate τ at the end. In the intersecting-surfaces approach, look for equations that decouple, such as du/dx depending only on x and u, or for combinations of A, B and C that sum to zero. A released verification problem asks for this second approach by name on a population model, so practise saying which approach you use.
Finally, keep a list of the characteristic families you meet, lines, parabolas, circles and hyperbolas, since recognising them quickly speeds up domain questions in Week 8. After each solution, sketch two or three characteristic curves and confirm that your arbitrary function is constant, or follows its ODE, along each of them.
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