# s3.16.1 Find the general solution to the system of differential equations

#### bigwave

Find the general solution to the system of differential equations

$$\begin{cases} y'_1&=2y_1+y_2-y_3 \\y'_2&=3y_2+y_3\\y'_3&=3y_3 \end{cases}$$

ok I assume the next step is

$$\begin{bmatrix} y'_1 \\y'_2 \\y'_3 \end{bmatrix}=\begin{bmatrix} 2 & 1 & -1 \\0 & 3 & 1 \\ 0 & 0 & 3 \end{bmatrix} \begin{bmatrix} y_1 \\y_2 \\y_3 \end{bmatrix}$$

I had this posted on another forum but think I was not starting out right so got no reply's

#### HallsofIvy

MHF Helper
I would not use matrices at all. The third equation, $$\displaystyle y_3'= 3y_3$$, involves only $$\displaystyle y_3$$ and can be integrated directly: $$\displaystyle y_3(t)= c_3e^{3t}$$.

The second equation is $$\displaystyle y_2'= 3y_2+ y_3= 3y_2+ c_3e^{3t}$$. That is a "non-homogeneous equation with constant equations". We can first solve the "associated homogeneous equation", $$\displaystyle y_2'= 3y_2$$. The general solution to that is $$\displaystyle y= c_2e^{3t}$$ just as before. We also seek a single solution to the entire equation. Normally, since the "non-homogeneous" part is $$\displaystyle c_3e^{3t}$$ we would try a multiple of that but that is already a solution to the homogeneous equation so we try $$\displaystyle y= Ac_3te^{3t}$$. Then $$\displaystyle y'= Ac_3e^{3t}+ 3Ac_3te^{3t}$$ so the equation becomes $$\displaystyle Ac_3e^{3t}+ 3Ac_3te^{3t}= Ac_3te^{3t}+ c_3e^{3t}$$. That reduces to $$\displaystyle Ac_3e^{3t}= c_3e^{3t}$$ so A= 1.
$$\displaystyle y_2= c_2e^{3t}+ c_3te^{3t}$$.

Now, the first equation is $$\displaystyle y_1'= 2y_1+ y_2- y_3= 2y_1+ c_2e^{3t}+ 2c_3te^{3t}- c_3e^{3t}$$. The associated homogeneous equation is $$\displaystyle y_1'= 2y_1$$ which has general solution $$\displaystyle y= c_1e^{2t}$$. To find a solution to the entire equation try $$\displaystyle y= (Ax+ B)e^{3t}$$ and find values for A and B that make that true.

1 person

#### romsek

MHF Helper
I would not use matrices at all.
But I suspect the point of the exercise is to know how to deal with repeated eigenvalues and rank deficient eigenvectors.

2 people