1. **State the problem:** Solve the differential equation $$|z| = r (\cos \theta + z \sin \theta)$$ where $z$ is the variable and $r$, $\theta$ are constants. 2. **Rewrite the equation:** The equation involves the modulus of $z$, which suggests $z$ might be complex. Let $z = x + iy$ where $x,y \in \mathbb{R}$. 3. **Express modulus:** We know $$|z| = \sqrt{x^2 + y^2}$$ 4. **Rewrite right side:** The right side is $$r (\cos \theta + z \sin \theta) = r \cos \theta + r z \sin \theta = r \cos \theta + r (x + iy) \sin \theta$$ 5. **Equate real and imaginary parts:** Since the left side $|z|$ is real, the imaginary part on the right must be zero for equality to hold. Imaginary part: $$r y \sin \theta = 0$$ 6. **Solve imaginary part:** This implies either $$y = 0$$ or $$\sin \theta = 0$$. 7. **Case 1: $y=0$ (real $z$):** Then $z = x$ real, and $$|z| = |x| = r (\cos \theta + x \sin \theta)$$ 8. **Solve for $x$:** If $x \geq 0$, then $|x| = x$, so $$x = r (\cos \theta + x \sin \theta)$$ Rearranged: $$x - r x \sin \theta = r \cos \theta$$ $$x (1 - r \sin \theta) = r \cos \theta$$ $$x = \frac{r \cos \theta}{1 - r \sin \theta}$$ If $x < 0$, then $|x| = -x$, so $$-x = r (\cos \theta + x \sin \theta)$$ $$-x = r \cos \theta + r x \sin \theta$$ $$-x - r x \sin \theta = r \cos \theta$$ $$x (-1 - r \sin \theta) = r \cos \theta$$ $$x = \frac{r \cos \theta}{-1 - r \sin \theta}$$ 9. **Case 2: $\sin \theta = 0$:** Then $\theta = k \pi$, $k \in \mathbb{Z}$. The equation reduces to: $$|z| = r \cos \theta$$ Since $|z| \geq 0$, this implies $r \cos \theta \geq 0$. Then $z$ can be any complex number with modulus $r \cos \theta$. 10. **Summary:** - If $\sin \theta \neq 0$, then $y=0$ and $x$ is given by the formulas above depending on sign. - If $\sin \theta = 0$, then $|z| = r \cos \theta$ and $z$ lies on the circle of radius $r \cos \theta$. **Final answer:** $$\boxed{\text{For } \sin \theta \neq 0, z = x = \frac{r \cos \theta}{1 - r \sin \theta} \text{ or } \frac{r \cos \theta}{-1 - r \sin \theta} \text{ with } y=0}$$ $$\boxed{\text{For } \sin \theta = 0, |z| = r \cos \theta \text{ and } z \in \{w \in \mathbb{C} : |w| = r \cos \theta\}}$$