Integral surface passing through a given curve

Now suppose that we want to find the integral surface of a given quasi-semi linear PDE

$$a(x,y,u)u_x+b(x,y,u)u_y=c(x,y,u),$$ (1.11)

containing a given curve $C$ described by the parameter equation

$$x=x(t),~y=y(t),~ ~u=u(t).$$ (1.12)

Solving the auxiliary equation, one gets the general solution

$$F(\psi_1,\psi_2)=0,$$

where $\psi_1(x,y,u)=c_1$ and $\psi_2(x,y,u)=c_2$ are integral curves obtained from the auxiliary equation. Using the parameter representations, one gets

$$\psi_1(x(t),y(t),u(t))=c_1,$$

$$\psi_2(x(t),y(t),u(t))=c_2.$$

Eliminate $t$ form this two relation to obtain the relation of the type

$$F(c_1,c_2)=0,$$

which gives the particular solution of Eq.() containing given curve Eq.().

Example 1.4.1   Find the integral surface of

$$x(y^2+u)u_x-y(x^2+u)u_y=(x^2-y^2)u,$$

containing the straight line $x+y$ and $z=1.$

Solution From the earlier illustration, we know the integral curves

$$\psi_1(x,y,u)=x^2+y^2-2u=c_1,$$

$$\psi_2(x,y,u)=xyu=c_2.$$

Parameters in the given curves as

$$x(t)=t~y(t)=t~$$and$$~u(t)=1.$$

Therefore,

$$\begin{split} 2t^2-2&=c_1,\\ \implies~-t^2&=c_2,\nonumber \end{split}$$

Therefore,

$$2c_2+c_1=-2,$$

and the particular solution is

$$2(xyu)+(x^2+y^2-2u)=-1.$$