# solving system of ODEs in Mathematica

I am using the DSolve to solve the following 2 ODEs $$\begin{cases} &\dot{y}(t)=z(t)-y(t)-\frac{c_1c_2y(t)}{-c_1c_2e^{-t}t+c_1(1-c_2)+c_3e^{-t}}\\ &\dot{z}(t)=-z(t)+\frac{c_1c_2y(t)}{-c_1c_2e^{-t}t+c_1(1-c_2)+c_3e^{-t}} \end{cases}$$ with initial value $z(0)=0$ and $y(0)\in(0,1)$.

Then I use the following code

system = {y'[t] == -y[t] + z[t] -
c1*c2*y[t]/(c1*(1 - c2) - c1*c2*Exp[-t]*t + c3*Exp[-t]),
z'[t] == -z[t] +
c1*c2*y[t]/(c1*(1 - c2) - c1*c2*Exp[-t]*t + c3*Exp[-t]),
z[0] == 0};


and

DSolve[system, {y[t], z[t]}, t]


However, Mathematica did nothing but beautify my input. So does this mean, Mathematica cannot solve this ODE in closed-form?

• You can format inline code and code blocks by selecting the code and clicking the {} button above the edit window. The edit window help button ? is also useful for learning how to format your questions and answers. You may also find this this meta Q&A helpful – Michael E2 Apr 12 '16 at 22:21
• Yeah, it probably means DSolve doesn't know how to solve it. – Michael E2 Apr 12 '16 at 22:32
• @MichaelE2 then is there any more powerful tool to solve ODE symbolically? – KevinKim Apr 13 '16 at 1:01
• No, I don't think there is. Sometimes one can apply a change of variables that converts the system to a form DSolve can recognize, but it's rare. I have nothing to suggest in that regard. Perhaps someone will be able to suggest something. – Michael E2 Apr 13 '16 at 3:10