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## 3. First-Order Nonlinear Partial Differential Equations

Preliminary remarks. For first-order partial differential equations in two independent variables, an exact solution

(*)    w = Φ(x, y, C1, C2)

that depends on two arbitrary constants C1 and C2 is called a complete integral. The general integral (general solution) can be represented in parametric form by using the complete integral (*) and the two equations

C2 = f(C1),

ΦC1 + ΦC2f′(C1) = 0,

where f(C1) is an arbitrary function, the prime stands for the derivative, and ΦC1 and ΦC2 are partial derivatives.

References

1. E. Kamke, Differentialgleichungen: Losungsmethoden und Losungen, II, Partielle Differentialgleichungen Erster Ordnung fur eine gesuchte Funktion, Akad. Verlagsgesellschaft Geest & Portig, Leipzig, 1965.
2. A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002.

### 3.3. Equations with Arbitrary Nonlinearities in Derivatives

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