1.3 Linear equations in two variables
Consider \begin {equation} \label {eq:linear1st} a(x, y)u_x + b(x, y)u_y + c(x, y)u = f(x, y) , \end {equation} with \(a\), \(b\), \(c\), \(f\) continuously differentiable and \((a, b) \neq (0, 0)\). The left side’s first two terms are the derivative of \(u\) in the direction of the vector field \((a, b)\).
Definition 1.3.1. The characteristic curves of (1.1) are the integral curves of the vector field \((a, b)\): \[ \frac {dx}{ds} = a(x, y), \qquad \frac {dy}{ds} = b(x, y) . \]
Theorem 1.3.2. Along a characteristic curve \((x(s), y(s))\), a \(C^1\) solution of (1.1) satisfies the ordinary differential equation \[ \frac {d}{ds}u\bigl (x(s), y(s)\bigr ) = f - cu . \]
Proof. By the chain rule, \(\frac {d}{ds}u(x(s), y(s)) = u_xx' + u_yy' = au_x + bu_y = f - cu\). □
The PDE has become a family of ODEs, one along each characteristic. To solve (1.1), one finds the characteristic curves, solves the ODE along each, and pieces the solutions together. If the characteristics are the level curves of a function \(\xi (x, y)\), so that \(a\xi _x + b\xi _y = 0\), then in the coordinates \(\xi \) and \(\eta = y\) (or any independent second coordinate) the equation becomes an ODE in \(\eta \) with \(\xi \) as a parameter, and the general solution contains an arbitrary function of \(\xi \).
Example 1.3.3. For \(xu_x + yu_y = 2u\) on \(x > 0\), the characteristics solve \(x' = x\), \(y' = y\): rays from the origin, the level curves of \(\xi = y/x\). Along a ray \(x = x_0e^s\), \(y = y_0e^s\), \(du/ds = 2u\), so \(u = u_0e^{2s}\), which grows like \(x^2\). The general solution is \(u = x^2F(y/x)\) for an arbitrary \(C^1\) function \(F\): the solutions are exactly the functions homogeneous of degree \(2\). (This is Euler’s theorem on homogeneous functions, read backwards.)
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