# How can apply a perturbation method in a system of equations?

If a have a system of equation

$$\frac{d u}{dt} = 1- u e^{\epsilon(q-1)}$$

$$\frac{d q}{dt} = u e^{\epsilon(q-1)}-q$$

$$u(0)=q(0)= 0$$

How Can I apply the perturbation methot

$$\epsilon$$ small parameter

• Expand the exponentials with a power series. – Paul Oct 17 at 20:46
• Ok, but when I solve the equations one by one. Or one equation have relation whit the other. $$u(t)=u_0(t)+\epsilonu_1(t)$$ I know the power series of $e^{\epsilon(q-1)}$ is $1 + \epsilon(q-1)+ 0.5\epsilon^2(q-1)^2$ when I introduce the previous power series in the first equation $$u'_0+u_0-1 +\epsilon (u'_1+u_0(q-1)+u_1)+ \epsilon^2 (u_0(q-1)^2+u_0(q-1)+u_1)+ O(\epsilon^3)=0$$ I can solve this but have some relation whit the other equation? – Conan Oct 17 at 21:17
• First introduce the new variable $q-1$. – John B Oct 18 at 8:35
• Do the same thing with $q$ (or $q-1$), write $q=q_0+\epsilon q_1+\dots$ and substitute in. – David Oct 18 at 17:09