UPSC 2019 Maths Optional Paper 2 Q7c — Step-by-Step Solution

20 marks · Section B

Classification and reduction to canonical form · PDEs · asked 8× in 13 yrs · Read the full method →

Question

Reduce the following second order partial differential equation to canonical form and find the general solution:

\frac{\partial^2 u}{\partial x^2}-2x\frac{\partial^2 u}{\partial x\partial y}+x^2\frac{\partial^2 u}{\partial y^2}=\frac{\partial u}{\partial y}+12x.

Technique

Classify via $B^2-AC=0$ (parabolic); one characteristic family $y+x^2/2=$ const gives $\xi$ ; choose $\eta=x$ ; transform; reduce to $u_{\eta\eta}=12\eta$ ; integrate twice.

Solution

Setup. Compare with $A u_{xx}+2B u_{xy}+C u_{yy}+\cdots=0$ :

A=1,\qquad 2B=-2x\Rightarrow B=-x,\qquad C=x^2.

Step 1 — Classify

Discriminant:

B^2-AC=(-x)^2-(1)(x^2)=x^2-x^2=0.

The equation is parabolic everywhere (one repeated family of characteristics).

Step 2 — Characteristic curves

Characteristic ODE $A\,dy^2-2B\,dx\,dy+C\,dx^2=0$ , i.e. $dy^2+2x\,dx\,dy+x^2dx^2=0$ , $(dy+x\,dx)^2=0$ :

\frac{dy}{dx}=-x\ \Longrightarrow\ y+\frac{x^2}{2}=\text{const}.

Take the new (characteristic) variable

\xi=y+\frac{x^2}{2},

and any independent second variable, simplest $\eta=x$ .

Step 3 — Transform the derivatives

With $\xi=y+\frac{x^2}{2},\ \eta=x$ : $\xi_x=x,\ \xi_y=1,\ \eta_x=1,\ \eta_y=0$ .

u_x=x\,u_\xi+u_\eta,\qquad u_y=u_\xi.

u_{yy}=u_{\xi\xi},\qquad u_{xy}=x\,u_{\xi\xi}+u_{\xi\eta},

u_{xx}=x^2u_{\xi\xi}+2x\,u_{\xi\eta}+u_{\eta\eta}+u_\xi.

Substitute into the LHS $u_{xx}-2x\,u_{xy}+x^2u_{yy}$ :

\big(x^2u_{\xi\xi}+2xu_{\xi\eta}+u_{\eta\eta}+u_\xi\big)-2x\big(xu_{\xi\xi}+u_{\xi\eta}\big)+x^2u_{\xi\xi}.

The $u_{\xi\xi}$ terms: $x^2-2x^2+x^2=0$ ; the $u_{\xi\eta}$ terms: $2x-2x=0$ . Left with $u_{\eta\eta}+u_\xi$ . The RHS is $u_y+12x=u_\xi+12x=u_\xi+12\eta$ . So

u_{\eta\eta}+u_\xi=u_\xi+12\eta\ \Longrightarrow\ u_{\eta\eta}=12\eta.

\boxed{\;\text{Canonical form: }\ \dfrac{\partial^2 u}{\partial\eta^2}=12\eta,\qquad \xi=y+\tfrac{x^2}{2},\ \eta=x.\;}

Step 4 — General solution

Integrate twice in $\eta$ (treating $\xi$ as a parameter):

u_\eta=6\eta^2+f(\xi),\qquad u=2\eta^3+\eta\,f(\xi)+g(\xi),

with $f,g$ arbitrary functions. Return to $(x,y)$ via $\eta=x,\ \xi=y+\frac{x^2}{2}$ :

Answer

\boxed{\;u(x,y)=2x^3+x\,f\!\Big(y+\tfrac{x^2}{2}\Big)+g\!\Big(y+\tfrac{x^2}{2}\Big).\;}