The graph integrates dx/ds = a, dy/ds = b, and du/ds = c with fourth-order Runge–Kutta steps.
Enter an explicit candidate u(x,y). The calculator estimates partial derivatives with centered finite differences and evaluates the PDE residual.
Candidate solution surface
A first-order quasilinear equation is written as a(x,y,u)uₓ + b(x,y,u)uᵧ = c(x,y,u). Its characteristic curves satisfy the coupled ordinary differential equations below.
Two independent first integrals I₁ and I₂ produce the general relation F(I₁,I₂) = 0, or I₂ = Φ(I₁).
- Enter the three coefficient expressions in the standard quasilinear form.
- Select an initial condition or define a parameterized initial curve.
- Choose precision, simplification, and numerical tracing settings.
- Press Solve PDE for symbolic pattern analysis and full steps.
- Press Trace characteristics for a numerical graph.
- Use candidate verification to test an explicit proposed solution.
- Copy, export, print, or save the displayed work as needed.
| PDE | Characteristic invariants | General solution |
|---|---|---|
| uₓ + uᵧ = 0 | x - y, u | u = Φ(x - y) |
| yuₓ + xuᵧ = 0 | x² - y², u | u = Φ(x² - y²) |
| uₓ + uuᵧ = 0 | u, y - xu | u = Φ(y - xu) |
| xuₓ + yuᵧ = u | y/x, u/x | u = xΦ(y/x) |
| uuₓ + uᵧ = 1 | u-y, x-yu+y²/2 | F(u-y, x-yu+y²/2)=0 |
| xuₓ + yuᵧ = u² | y/x, -1/u-ln|x| | -1/u-ln|x|=Φ(y/x) |
Quasilinear equations and characteristic curves
A quasilinear PDE is linear in its highest derivatives. Its coefficients may still depend on the unknown function. Characteristics convert the PDE into linked ordinary differential equations.
The characteristic parameter tracks motion through three-dimensional x, y, and u space. Each curve carries solution data from an initial set. This geometric viewpoint makes many first-order equations manageable.
Characteristic invariants
An invariant remains constant along every characteristic trajectory. Two independent invariants usually define the complete integral. Their functional relationship introduces the arbitrary function in general solutions.
Some equations yield invariants through direct pairwise integration. Others require multipliers, substitutions, or exact differential combinations. Highly nonlinear systems may only permit numerical characteristic tracing.
Initial curves and uniqueness
Cauchy data are commonly prescribed on a parameterized initial curve. The curve should cut across characteristic directions rather than follow them. This requirement is called the non-characteristic condition.
When the determinant vanishes, local uniqueness can fail. Incompatible data may also prevent any smooth solution. Intersecting characteristics can create steep gradients and shock formation.
Symbolic and numerical limits
This page recognizes several important characteristic patterns automatically. It also constructs the correct general ODE system for other inputs. Numerical RK4 tracing then reveals characteristic geometry and solution transport.
Numerical verification estimates derivatives at a selected point. A small residual supports the proposed explicit solution. It does not replace a rigorous symbolic proof everywhere.
No saved calculations appear in this session yet.
What is a quasilinear first-order PDE?
It is linear in first derivatives, although coefficients may depend on x, y, and u.
What does the method of characteristics do?
It transforms the PDE into ordinary differential equations along special curves.
Why does the answer contain an arbitrary function?
A first-order PDE generally admits a family of solutions parameterized by one function.
Can this calculator solve every quasilinear PDE symbolically?
No finite pattern engine solves every possible symbolic characteristic system. Numerical tracing remains available.
What is a characteristic invariant?
It is an expression whose derivative vanishes along each characteristic curve.
What is a non-characteristic initial curve?
It is an initial curve that crosses characteristic directions and supports local uniqueness.
Why can shocks appear?
Different characteristics may intersect, making a smooth single-valued solution impossible.
How is the numerical graph calculated?
The graph uses fourth-order Runge–Kutta integration from sampled initial points.
How does candidate verification work?
It estimates uₓ and uᵧ numerically, then computes the PDE residual.