Solving two first order differential equations in four variables by adding them together

Question

If

$$ \displaystyle \frac{dy_{1}}{dx} = -3y_{1} + y_{2} $$ $$ \displaystyle \frac{dy_{2}}{dx} = y_{1} - 3y_{2} $$

Find $y_{1}\left(t\right)$ and $y_{2}\left(t\right)$, given that $y_{1}\left(0\right) = 2$ and $y_{2}\left(0\right) = 0$

My Attempt

Adding the two differential equations gives: $ \displaystyle \frac{dy_{2}}{dx} + \displaystyle \frac{dy_{1}}{dx} = -3y_{1} + y_{2} + y_{1} - 3y_{2} $

$$ \implies \displaystyle \frac{d\left(y_{1} + y_{2} \right)}{dx} + 2\left(y_{1} + y_{2}\right) = 0$$

$$ \implies \left(y_{1} + y_{2} \right) = \lambda\exp(-2t)\ \ \ \lambda \in \mathbb{R}$$

Which I then resubstituted into the initial differential equations to solve for $y_{1}$ and $y_{2}$, yielding: $$y_{1} = \exp(-2t) + \exp(-4t)$$ $$y_{2} = \exp(-2t) - \exp(-4t)$$

Question

Is this a perfectly valid approach to solving these kinds of problems? And is there a better method which isn't dependent on the coefficients lining up nicely when summing the equations together?

Answer 1

Instead of substituting back, you could also apply the same method to the difference of both equations.

As it is, you have intuitively found one of the eigenvectors of the system matrix. In that sense you are well inside the standard solution method. (You could have used $y_1(0)+y_2(0)=2$ to find $λ=2$ directly.)

But be aware that in general it is not that easy to guess those eigenvectors and reduce the system to uncoupled order $1$ scalar equations.