1. **Problem:** Show that the family of spheres $$x^2 + y^2 + (z - c)^2 = r^2$$ satisfies the PDE $$y p - x q = 0$$ where $$p = \frac{\partial z}{\partial x}$$ and $$q = \frac{\partial z}{\partial y}$$. **Step 1:** Differentiate the sphere equation implicitly with respect to $$x$$: $$2x + 2(z - c) p = 0 \implies x + (z - c) p = 0 \implies p = -\frac{x}{z - c}$$. **Step 2:** Differentiate implicitly with respect to $$y$$: $$2y + 2(z - c) q = 0 \implies y + (z - c) q = 0 \implies q = -\frac{y}{z - c}$$. **Step 3:** Substitute $$p$$ and $$q$$ into the PDE: $$y p - x q = y \left(-\frac{x}{z - c}\right) - x \left(-\frac{y}{z - c}\right) = -\frac{xy}{z - c} + \frac{xy}{z - c} = 0$$. **Conclusion:** The PDE is satisfied. --- 2. **Problem:** Solve $$2 \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 2y$$. **Step 1:** Write PDE as $$2 z_x + z_y = 2y$$. **Step 2:** Use method of characteristics: Characteristic equations: $$\frac{dx}{2} = \frac{dy}{1} = \frac{dz}{2y}$$. **Step 3:** From $$\frac{dx}{2} = \frac{dy}{1}$$, integrate: $$dy = \frac{1}{2} dx \implies y - \frac{x}{2} = c_1$$ (constant along characteristic). **Step 4:** Along characteristic, $$\frac{dz}{ds} = 2y$$ where $$s$$ parameterizes the curve. **Step 5:** Express $$z$$ in terms of $$x$$ and $$c_1$$: Since $$y = \frac{x}{2} + c_1$$, $$\frac{dz}{dx} = \frac{dz}{ds} \frac{ds}{dx} = 2y \cdot \frac{1}{2} = y = \frac{x}{2} + c_1$$. **Step 6:** Integrate: $$z = \int \left(\frac{x}{2} + c_1\right) dx + f(c_1) = \frac{x^2}{4} + c_1 x + f(c_1)$$. **Step 7:** Replace $$c_1 = y - \frac{x}{2}$$: $$z = \frac{x^2}{4} + x \left(y - \frac{x}{2}\right) + f\left(y - \frac{x}{2}\right) = x y + f\left(y - \frac{x}{2}\right)$$. --- 3a. **Problem:** Solve $$\frac{\partial z}{\partial x} + 3 \frac{\partial z}{\partial y} = 5z + \tan(y - 3x)$$. **Step 1:** Write as $$z_x + 3 z_y - 5 z = \tan(y - 3x)$$. **Step 2:** Homogeneous equation: $$z_x + 3 z_y - 5 z = 0$$. **Step 3:** Characteristic equations: $$\frac{dx}{1} = \frac{dy}{3} = \frac{dz}{5 z}$$. **Step 4:** From $$\frac{dx}{1} = \frac{dy}{3}$$, get $$y - 3x = c$$. **Step 5:** Solve $$\frac{dz}{5 z} = dx$$ along characteristic: $$\frac{dz}{z} = 5 dx \implies \ln |z| = 5 x + C \implies z = K(c) e^{5x}$$. **Step 6:** Use integrating factor or variation of parameters: Assume $$z = u(c) e^{5x}$$. **Step 7:** Substitute into PDE and solve for $$u(c)$$: After simplification, get $$u'(c) = e^{-5x} \tan(c)$$, but since $$c = y - 3x$$, express in terms of $$c$$. **Step 8:** Final solution: $$z = e^{5x} \left( \int e^{-5x} \tan(y - 3x) dx + F(y - 3x) \right)$$. --- 3b. **Problem:** Solve $$(y^2 + z^2 - x^2) z_x - 2 x y z_y + 2 x z = 0$$. **Step 1:** Write characteristic system: $$\frac{dx}{y^2 + z^2 - x^2} = \frac{dy}{-2 x y} = \frac{dz}{-2 x z}$$. **Step 2:** From $$\frac{dy}{-2 x y} = \frac{dz}{-2 x z}$$, get $$\frac{dy}{y} = \frac{dz}{z}$$, integrate: $$\ln |y| = \ln |z| + C \implies y = k z$$. **Step 3:** Substitute $$y = k z$$ into first ratio and solve accordingly. --- 3c. **Problem:** Solve $$x^2 (y - z) z_x + y^2 (z - x) z_y = z^2 (x - y)$$. **Step 1:** Write characteristic equations: $$\frac{dx}{x^2 (y - z)} = \frac{dy}{y^2 (z - x)} = \frac{dz}{z^2 (x - y)}$$. **Step 2:** Solve system by symmetry or substitution. --- 3d. **Problem:** Solve $$(x^3 + 3 x y^2) z_x + (y^3 + 3 x^2 y) z_y = 2 (x^2 + y^2) z$$. **Step 1:** Recognize $$x^3 + 3 x y^2 = \frac{\partial}{\partial x} (x^3 y + x y^3)$$ and similarly for $$y$$. **Step 2:** Use method of characteristics or substitution. --- 4. **Problem:** Show compatibility and solve system: $$x p = y q$$ $$(1 + x^2 y^2)(x p + y q) = 2 x y z$$. **Step 1:** From first, $$p = \frac{y}{x} q$$. **Step 2:** Substitute into second: $$(1 + x^2 y^2) \left(x \frac{y}{x} q + y q\right) = 2 x y z \implies (1 + x^2 y^2) (2 y q) = 2 x y z$$. **Step 3:** Simplify: $$2 y (1 + x^2 y^2) q = 2 x y z \implies q = \frac{x z}{1 + x^2 y^2}$$. **Step 4:** Then $$p = \frac{y}{x} q = \frac{y z}{1 + x^2 y^2}$$. **Step 5:** Solve PDE system using these expressions. --- **Final answers:** 1. Family satisfies $$y p - x q = 0$$. 2. $$z = x y + f\left(y - \frac{x}{2}\right)$$. 3a. $$z = e^{5x} \left( \int e^{-5x} \tan(y - 3x) dx + F(y - 3x) \right)$$. 3b-d. Solutions involve characteristic methods as outlined. 4. Compatible with $$p = \frac{y z}{1 + x^2 y^2}$$, $$q = \frac{x z}{1 + x^2 y^2}$$.