Background: I've solved this linear PDE: $$\frac{\partial H_a}{\partial p_r}-g_0({p_r})\frac{\partial H_a}{\partial z}=0$$ Like this (and this is true) that I used method of characteristics: $$\frac{dp_r}{ds}=1 \And (p_r(0)=0)\implies p_r=s$$ $$\frac{dz}{ds}=g_0({p_r}) \And(z(0)=\aleph)\implies z=-\int_{0}^{p_r} g_0(s) ds+\aleph$$ $$\frac{dH_a}{ds}=0 \And (H_a(0)=\phi(\aleph))\implies H_a(z,p_r)=\phi(z+\int_{0}^{p_r} g_0(s) ds)$$
Question: Now, I should solve this quasilinear PDE:
$$\frac{\partial H_a}{\partial p_r}-2g_0({p_r})\frac{\partial H_a}{\partial z}=g_0({p_r})k_0z$$
Although I like to use method of characteristics again, I don't know hot to do. So I do this:
Forming this: $$\frac{dp_r}{1}=\frac{dz}{-2g_0({p_r})}=\frac{dH_a}{g_0({p_r})k_0z}$$ Taking first two fractions integral leads: $$dz=-2g_0({p_r})dp_r \implies z=-2\int_{0}^{p_r} g_0(s) ds+c_1$$ And for last two fractions: $$\frac{k_0}{-2}zdz=dH_a \implies H_a=\frac{k_0}{-2}z^2+c_2$$
And I'm sure I miss some points (for example $c_1$ and $c_2$ are not necessarily constants). Could anyone help me?
Without checking the preliminary calculus above.
You obtained the equations of two families of characteristic curves : $$z+2\int_{0}^{p_r} g_0(s) ds=c_1$$ $$ H_a-\frac{k_0}{-2}z^2=c_2$$ Those equations are valid on the respective characteristic curves for any $c_1$ and $c_2$. But this is not true everywhere on the surface linking the characteristic curves : $c_1$ and $c_2$ are no longer independent. The relationship can be expressed on the form of an implicit equation : $\Phi(c_1,c_2)=0$. This leads to general solution of the PDE on the form : $$\Phi\left((z+2\int_{0}^{p_r} g_0(s) ds)\:,\:(H_a-\frac{k_0}{-2}z^2)\right)=0$$ where $\Phi$ is any differentiable function of two variables.
An equivalent way to express the above relationship consists in expressing one variable as a function of the other : $c_2=F(c_1)$ or $c_1=G(c_2)$ where $F$ and $G$ are any differentiable functions, but related (one is the inverse of the other).
For example with the first : $$\left(H_a-\frac{k_0}{-2}z^2\right)=F\left(z+2\int_{0}^{p_r} g_0(s) ds\right)$$ So, an explicit form of the general solution is : $$H_a=-\frac{k_0}{2}z^2+F\left(z+2\int_{0}^{p_r} g_0(s) ds\right)$$