Discussion
The following partial differential equation :
\[ \begin{aligned} &(x^{2} + z^{2})\frac{\partial z}{\partial x} - xy\frac{\partial z}{\partial y} \\ &= z^{3}x + y^{2} \end{aligned} \]
is :
Answer: D
The equation is quasi-linear.
A first-order equation \(a\,z_{x} + b\,z_{y} = c\) is quasi-linear when it is linear in the derivatives \(z_{x}, z_{y}\) but the coefficients \(a, b\) may depend on \(z\) itself. Here the coefficient of \(z_{x}\) is \(x^{2} + z^{2}\), which contains \(z\), and the derivatives appear only to the first power.
It is not linear or semi-linear, because those need \(a\) and \(b\) to depend on \(x, y\) only. It is not fully non-linear, since there are no products or powers of the derivatives. The \(z^{3}\) on the right is allowed in a quasi-linear equation.
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