Question

Which of the following is not a homogeneous equation?

(a) \(\frac{∂^2 u}{∂t^2}-c^2 \frac{∂^2 u}{∂x^2}=0\)

(b) \(\frac{∂^2 u}{∂x^2}+\frac{∂^2 u}{∂y^2}=0\)

(c) \(\frac{∂^2 u}{∂x^2}+(\frac{∂^2 u}{∂x∂y})^2+\frac{∂^2 u}{∂y^2}=x^2+y^2\)

(d) \(\frac{∂u}{∂t} -T\frac{∂^2 u}{∂x^2}=0\)

The question was posed to me in an interview for internship.

My query is from Homogeneous Linear PDE with Constant Coefficient in section Partial Differential Equations of Engineering Mathematics

Answer 1

Right answer is (c) \(\frac{∂^2 u}{∂x^2}+(\frac{∂^2 u}{∂x∂y})^2+\frac{∂^2 u}{∂y^2}=x^2+y^2\)

To explain I would say: As we know that homogeneous equations are those in which the dependent variable (and its derivatives) appear in terms with degree exactly one, hence the equation,

\(\frac{∂^2 u}{∂x^2}+(\frac{∂^2 u}{∂x∂y})^2+\frac{∂^2 u}{∂y^2}=x^2+y^2\) is not a homogenous equation (since its degree is 2).