Differential equation non linear first order

Question

Can you find the solution of that equation please? I don't know how to solve it. I use it to solve a dynamic system in macroeconomics for my homework. $$\frac{dy}{dt} = y^3 + 1$$

Answer 1

Hint: You can use separation, and then integrate. Try that.

Answer 2

$$\int \frac{dy}{1+y^3}=\int dt$$ the only difficult part in ye question is to find $$\int \frac{dy}{1+y^3}$$ which is $$\int \frac{dy}{(1+y)^3-3y(1+y)}$$ substitute $1+y=t$. then the integral becomes $$\int \frac{dt}{t^3-3t(t-1)}$$ $$\int \frac{dt}{t(t^2-3t+3)}$$ now convert the integrand to partial fractions and evaluate. $$\frac{1}{t(t^2-3t+3)}=\frac{A}{t} +\frac{Bt+C}{(t^2-3t+3)}$$ find $A,B and C$.further evaluate.