Definitions — Question 1

Question 1 – Solution

Strategy. Count the highest derivative order, then examine how the unknown function and its derivatives occur. A linear equation has coefficients depending only on independent variables.

Step 1: Classification.

In (E), $u_t$ has order one and $u_{xx}$ has order two; the highest order is therefore two. In (C), squaring a first derivative does not make it a second derivative.

Step 2: Homogeneity. The two linear equations can be written as $$x^2y''+\sin(x)y'-3y=e^x,\qquad u_t-4u_{xx}-xu=0.$$ Thus (A) is nonhomogeneous, and (E) is homogeneous. Here “homogeneous” is being used in the linear-equation sense.

Step 3: Coefficients versus unknowns. The factors $x^2$, $\sin(x)$, and $x$ are specified functions of independent variables, so they are allowed coefficients. By contrast, $y$ multiplying $y'$ in (B) is part of the unknown solution.

Check. Order and linearity are different properties: (C) is first order but nonlinear, whereas (A) is second order but linear. The vanishing of $x^2$ at $0$ affects normalization of (A), not this linearity classification.