1. **Eliminate arbitrary constants a and b:** (a) Given $z = ax + by + a^2 b^2$. Step 1: Differentiate partially w.r.t $x$ and $y$: $$p = \frac{\partial z}{\partial x} = a, \quad q = \frac{\partial z}{\partial y} = b.$$ Step 2: Express $a$ and $b$ in terms of $p$ and $q$: $$a = p, \quad b = q.$$ Step 3: Substitute back into original equation: $$z = px + qy + p^2 q^2.$$ Step 4: Differentiate $z$ w.r.t $x$ and $y$ again and eliminate $p, q$ to get PDE. (b) Given $z = ax^2 + by^2$. Step 1: Differentiate partially: $$p = \frac{\partial z}{\partial x} = 2ax, \quad q = \frac{\partial z}{\partial y} = 2by.$$ Step 2: Express $a$ and $b$: $$a = \frac{p}{2x}, \quad b = \frac{q}{2y}.$$ Step 3: Substitute into $z$: $$z = x^2 \frac{p}{2x} + y^2 \frac{q}{2y} = \frac{xp}{2} + \frac{yq}{2}.$$ Step 4: Rearrange to get PDE. 2. **Eliminate arbitrary function:** (a) Given $z = f(x^2 + y^2)$. Step 1: Let $u = x^2 + y^2$, then $z = f(u)$. Step 2: Differentiate: $$p = \frac{\partial z}{\partial x} = f'(u) 2x, \quad q = \frac{\partial z}{\partial y} = f'(u) 2y.$$ Step 3: Eliminate $f'(u)$: $$\frac{p}{2x} = \frac{q}{2y} \implies \frac{p}{x} = \frac{q}{y}.$$ Step 4: PDE is: $$yp - xq = 0.$$ (b) Given $z = x f(y/x)$. Step 1: Let $v = y/x$, then $z = x f(v)$. Step 2: Differentiate: $$p = \frac{\partial z}{\partial x} = f(v) - v f'(v), \quad q = f'(v).$$ Step 3: Eliminate $f'(v)$: $$q = f'(v), \quad p = f(v) - v q.$$ Step 4: Rearranged PDE: $$x p + y q = z.$$ 3. **General solution of PDE:** $$(y + z)p + (z + x)q = x + y.$$ Step 1: Use method of characteristics or substitution to solve. 4. **Equation of surface cutting orthogonally:** Given family: $$2xz + 3yz = a(z + 2).$$ Step 1: Find gradient vector of family surfaces. Step 2: Orthogonal surface satisfies dot product zero. Step 3: Use given circle condition to find particular solution. 5. **Integral surface cutting orthogonally:** Family: $$x^2 + y^2 = cz,$$ Curve: $$x = a \cos t, y = a \sin t, z = b.$$ Step 1: Find gradient of family. Step 2: Orthogonal surface satisfies PDE. Step 3: Use curve to find particular solution. 6. **Complete integral of PDE:** $$(p^2 + q^2)x = pz.$$ Step 1: Use method of complete integrals. 7. **General solution of PDE:** $$[2D^2 - DD' - (D')^2 + D - D'] z = e^{2x + 3y}.$$ Step 1: Identify operators $D, D'$ and solve accordingly. 8. **Particular integrals:** (a) $$\frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 z}{\partial y^2} = x^2 + y^2,$$ (b) $$\frac{\partial^2 z}{\partial x^2} + 2 \frac{\partial^2 z}{\partial x \partial y} + \frac{\partial^2 z}{\partial y^2} = 3x + 2y.$$ Step 1: Use method of undetermined coefficients. 9. **General solution:** $$(x + y^2)p + yq = z + x^2.$$ Step 1: Solve using characteristic method. 10. **Classify PDEs:** (a) $u_{xx} + u_{yy} = 0$ elliptic. (b) $u_{tt} - c^2 u_{xx} = 0$ hyperbolic. (c) $u_{xx} - 4u_{xy} + 4u_{yy} = 0$ parabolic. 11. **Solve heat equation:** $$u_t = k u_{xx}, 0 < x < L, t > 0,$$ with boundary and initial conditions. Step 1: Use separation of variables. 12. **Solve wave equation:** $$u_{tt} = c^2 u_{xx}, 0 < x < L, t > 0,$$ with given conditions. Step 1: Use d'Alembert or separation of variables. 13. **Solve Laplace's equation:** $$u_{xx} + u_{yy} = 0, 0 < x < a, 0 < y < b,$$ with boundary conditions. Step 1: Use separation of variables and Fourier series. Final answers depend on detailed solving steps for each problem.