Find the general integral of the equation s + yq + z = 0, s…

Find the general integral of the equation \[ s + yq + z = 0, \qquad s = \frac{\partial^2 z}{\partial x\,\partial y},\quad q = \frac{\partial z}{\partial y}. \]

Step 1 · Recognise the exact form

Write the equation as \[ \frac{\partial^2 z}{\partial x\,\partial y} + y\,\frac{\partial z}{\partial y} + z = 0. \] Since $\dfrac{\partial}{\partial y}(yz) = z + y\dfrac{\partial z}{\partial y}$, the equation becomes \[ \frac{\partial}{\partial y}\!\left(\frac{\partial z}{\partial x} + yz\right) = 0. \]

Step 2 · Integrate with respect to $y$

Since $\dfrac{\partial}{\partial y}\!\left(p + yz\right) = 0$, the expression $p+yz$ is independent of $y$, hence a function of $x$ alone: \[ \frac{\partial z}{\partial x} + yz = f(x). \tag{2} \] This is a first-order linear ODE in $z$ with respect to $y$, with $x$ as a parameter and $f(x)$ an arbitrary function of $x$.

Step 3 · Solve the ODE (2)

The integrating factor is $e^{\int y\,dy} = e^{y^2/2}$. Multiply (2) by $e^{y^2/2}$: \[ \frac{\partial}{\partial y}\!\left(z\,e^{y^2/2}\right) = f(x)\,e^{y^2/2}. \] Integrate with respect to $y$: \[ z\,e^{y^2/2} = f(x)\int e^{y^2/2}\,dy + \varphi(x), \] where $\varphi(x)$ is an arbitrary function of $x$. Dividing by $e^{y^2/2}$ and writing $\Psi(x)=f(x)$: \[ z = e^{-y^2/2}\!\left[\Psi(x)\int_0^y e^{t^2/2}\,dt + \Phi(x)\right]. \]

Verification

Compute $\partial z/\partial x$: \[ \frac{\partial z}{\partial x} = e^{-y^2/2}\!\left[\Psi'(x)\int_0^y e^{t^2/2}\,dt + \Phi'(x)\right] = f(x) - yz, \] which is exactly equation (2). Differentiating (2) with respect to $y$: \[ s + z + y\frac{\partial z}{\partial y} = 0 \;\Longrightarrow\; s + yq + z = 0. \checkmark \]