How do I start this System of DEs?

Question

So I'm not too sure on how I can start this problem without eigen method. Any help would be appreciated.

Answer 1

$\frac{dy_1}{dx}=3y_1$; $\frac{dy_1}{y_1}=3dx$; $\int{\frac{dy_1}{y_1}}=\int3dx$.
Thus, $ln(y_1)=3x+c$ and the solution for the first equation is $y_1=c_1e^{3x}$.
Now, $\frac{dy_2}{dx}=c_1e^{3x}+y_2$; $\frac{dy_2}{dx}-y_2=c_1e^{3x}$; $y_2=c_2e^x+\frac{c_1e^{3x}}{2}$; and I leave the last one for you to solve.

Answer 2

Solve the first one first $$y'_1=3y_1 \implies \ln(y_1)=3x+K \implies y_1=Ke^{3x}$$ Then solve the second equation and plug $y_1$ into the equation.. $$y'_2=y_1+y_2 \implies y'_2=y_2+Ke^{3x}$$ Solve that one and do the same for the third equation. $$y'_3=y_1+y_3 \implies y'_3=y_3+Ke^{3x}$$