91Ó°ÊÓ

In the method of characteristics, we introduce new variables, say \(p\) and \(q\), to represent the derivatives of \(u\): $$p = \frac{\partial u}{\partial x}, \quad q = \frac{\partial u}{\partial y}$$ With these new variables, we can rewrite the given PDE as: $$p = y q$$ Next, we need to set up the system of ODEs using the total derivatives with respect to a new variable \(r\): $$\frac{dx}{dr} = p' = \frac{\partial p}{\partial x}\frac{dx}{dr} + \frac{\partial p}{\partial y}\frac{dy}{dr}$$ $$\frac{dy}{dr} = q' = \frac{\partial q}{\partial x}\frac{dx}{dr} + \frac{\partial q}{\partial y}\frac{dy}{dr}$$ However, we must also consider the total derivatives of \(u\) with respect to \(x\) and \(y\). Since \(\frac{\partial x}{\partial r} = p'\) and \(\frac{\partial y}{\partial r} = q'\), we can rewrite the equations above as: $$p' = \frac{\partial p}{\partial x}p' + \frac{\partial p}{\partial y}q'$$ $$q' = \frac{\partial q}{\partial x}p' + \frac{\partial q}{\partial y}q'$$

Now we need to find the equations that help us connect the derivatives \(p\) and \(q\) with the coordinates \((x, y)\). Using the equation from the PDE: $$p = y q$$ Then, we take the total derivatives with respect to \(r\): $$\frac{dp}{dr} = y \frac{dq}{dr} + q \frac{dy}{dr}$$ Plugging in the expressions for \(p'\) and \(q'\), we obtain: $$p' = y q' + q q'$$ Comparing this with the original equation for \(p'\): $$p' = \frac{\partial p}{\partial x}p' + \frac{\partial p}{\partial y}q'$$ We have the following system of equations: $$\begin{cases} \frac{\partial p}{\partial x} = 0 \\ \frac{\partial p}{\partial y} = y + q \end{cases}$$ Similarly, for the equation for \(q'\): $$q' = \frac{\partial q}{\partial x}p' + \frac{\partial q}{\partial y}q'$$ Since \(p = yq\), we rewrite it as: $$q' = \frac{\partial q}{\partial x}yp + \frac{\partial q}{\partial y}q'$$ Comparing these, we have the following system of equations: $$\begin{cases} \frac{\partial q}{\partial x} = y \\ \frac{\partial q}{\partial y} = 0 \end{cases}$$

Solving the characteristic equations, we get: For \(p\): $$p = \int_{}^{} (\frac{\partial p}{\partial y}) dy = \int_{}^{} (y + q) dy = y^2/2 + qy + p_0(x)$$ For \(q\): $$q = \int_{}^{} (\frac{\partial q}{\partial x}) dx = \int_{}^{} (y) dx = xy + q_0(y)$$ Substituting \(q = xy + q_0(y)\) in \(p = y^2/2 + qy + p_0(x)\), we get: $$p = y^2 / 2 + (xy + q_0(y))y + p_0(x)$$ These are the characteristic equations for this given PDE.