IGNOU MTE 8 2026 - Verify that the Pfaffian differential equationis integrable

Question

Verify that the Pfaffian differential equation

$$yz \, dx + (x^2y - zx) \, dy + (x^2z - xy) \, dz = 0$

is integrable and hence find its integral.
b) Solve the following equation by Jacobi's method

$$x^2 \frac{\partial u}{\partial x} - \left( \frac{\partial u}{\partial y} \right)^2 - a \left( \frac{\partial u}{\partial z} \right)^2 = 0.$
c) Show that $2z = (ax + y)^2 + b$ , where a, b are arbitrary constants is a complete integral of $px + qy - q^2 = 0$ .

09 Jan 2026

Answer :

Word Count : 204

Consider the Pfaffian form (M,dx+N,dy+P,dz=0) with (M=yz,; N=x^{2}y-zx,; P=x^{2}z-xy). The condition for integrability is (M\left(\frac{\partial N}{\partial z}-\frac{\partial P}{\partial y}\right)+N\left(\frac{\partial P}{\partial x}-\frac{\partial M}{\partial z}\right)+P\left(\frac{\partial M}{\partial y}-\frac{\partial N}{\partial x}\right)=0). Now (\frac{\partial N}{\partial z}=-x,; \frac{\partial P}{\partial y}=-x), hence the first bracket is zero. (\frac{\partial P}{\partial x}=2xz-y,; \frac{\partial M}{\partial z}=y), so (\frac{\partial P}{\partial x}-\frac{\partial M}{\partial

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