1. **State the problem:** Find the integral curves of the system given by the equation $$\frac{dx}{x+z} = \frac{dy}{y} = \frac{dz}{z+y^2}$$. 2. **Understand the problem:** Integral curves are curves along which the given differential relations hold. Here, we have a system of three variables $x,y,z$ with their differentials related. 3. **Step 1: Set the common ratio to a parameter $dt$:** $$\frac{dx}{x+z} = \frac{dy}{y} = \frac{dz}{z+y^2} = dt$$ This gives three separate differential equations: $$dx = (x+z) dt$$ $$dy = y dt$$ $$dz = (z + y^2) dt$$ 4. **Step 2: Solve for $y$:** From $$dy = y dt$$, we have $$\frac{dy}{y} = dt$$ Integrate both sides: $$\int \frac{dy}{y} = \int dt \implies \ln|y| = t + C_1$$ Exponentiate: $$y = C e^{t}$$ where $C = e^{C_1}$. 5. **Step 3: Express $z$ in terms of $t$ and $C$:** From $$dz = (z + y^2) dt$$, substitute $y = C e^{t}$: $$dz = (z + C^2 e^{2t}) dt$$ Rewrite as: $$\frac{dz}{dt} - z = C^2 e^{2t}$$ 6. **Step 4: Solve the linear ODE for $z$:** The integrating factor is: $$\mu(t) = e^{-t}$$ Multiply both sides: $$e^{-t} \frac{dz}{dt} - e^{-t} z = C^2 e^{t}$$ Which simplifies to: $$\frac{d}{dt} (z e^{-t}) = C^2 e^{t}$$ Integrate both sides: $$z e^{-t} = C^2 \int e^{t} dt + K = C^2 e^{t} + K$$ Multiply both sides by $e^{t}$: $$z = C^2 e^{2t} + K e^{t}$$ 7. **Step 5: Express $x$ in terms of $t$:** From $$dx = (x + z) dt$$, rewrite as: $$\frac{dx}{dt} = x + z = x + C^2 e^{2t} + K e^{t}$$ Rewrite: $$\frac{dx}{dt} - x = C^2 e^{2t} + K e^{t}$$ 8. **Step 6: Solve the linear ODE for $x$:** Integrating factor: $$\mu(t) = e^{-t}$$ Multiply both sides: $$e^{-t} \frac{dx}{dt} - e^{-t} x = C^2 e^{t} + K$$ Which simplifies to: $$\frac{d}{dt} (x e^{-t}) = C^2 e^{t} + K$$ Integrate both sides: $$x e^{-t} = C^2 \int e^{t} dt + K t + M = C^2 e^{t} + K t + M$$ Multiply both sides by $e^{t}$: $$x = C^2 e^{2t} + K t e^{t} + M e^{t}$$ 9. **Step 7: Summary of solutions:** $$y = C e^{t}$$ $$z = C^2 e^{2t} + K e^{t}$$ $$x = C^2 e^{2t} + K t e^{t} + M e^{t}$$ where $C,K,M$ are constants determined by initial conditions. 10. **Interpretation:** These parametric equations describe the integral curves of the given system.