Question

What is the general form of second order non-linear partial differential equations (x and y being independent variables and z being a dependent variable)?

(a) \(F(x,y,z,\frac{∂z}{∂x},\frac{∂z}{∂y},\frac{∂^2 z}{∂x^2},\frac{∂^2 z}{∂y^2},\frac{∂^2 z}{∂x∂y})=0\)

(b) \(F(x,z,\frac{∂z}{∂x},\frac{∂z}{∂y},\frac{∂^2 z}{∂x^2},\frac{∂^2 z}{∂y^2})=0\)

(c) \(F(y,z,\frac{∂z}{∂x},\frac{∂z}{∂y})=0\)

(d) F(x,y)=0

I had been asked this question in my homework.

Enquiry is from Solution of Second Order P.D.E. topic in portion Partial Differential Equations of Engineering Mathematics

Answer 1

Right answer is (a) \(F(x,y,z,\frac{∂z}{∂x},\frac{∂z}{∂y},\frac{∂^2 z}{∂x^2},\frac{∂^2 z}{∂y^2},\frac{∂^2 z}{∂x∂y})=0\)

Explanation: The most general second order partial differential equation in two independent variables x and y, and z as the dependent variable has the form,

\(F(x,y,z,\frac{∂z}{∂x},\frac{∂z}{∂y},\frac{∂^2 z}{∂x^2},\frac{∂^2 z}{∂y^2},\frac{∂^2 z}{∂x∂y})=0\)