Finding the differential equation given certain solutions

Question

I'm stuck at this exercise of my notes:

Find the differential equation that has the solutions:

$$\phi_1= e^{2t} (13\cos{t}, −26\sin{ t}, −26 \sin {t})$$ $$\phi_2= 7e^{2t}(−2 \cos{t} − 3 \sin {t}, −6 \cos{t} + 4 \sin {t}, −6 \cos{t} + 4 \sin {t}) + 2e^{−3t}(7, 8, 34)$$ $$\phi_3=e^{2t}(\sin {t}, 2 \cos{t}, 2 \cos{t})$$

It's the first exercise on my notes like this. How do we usually attack this kind of exercises? What's the procedure to follow?

Answer 1

Note that the $\phi_i$ are functions of vectors in $3$ dimensions. Differentiate the $\phi_i$ wrt $t$ component-wise, possibly more than once. Then eliminate $t$ from these equations to get three equations in terms of the $\phi_i$ and their derivatives. Then by having $$\phi=(\phi_1,\phi_2,\phi_3),$$ and its derivatives defined similarly in terms of their components, you have the equation you seek in terms of $\phi$ and its derivatives.

NB. It will be messy, but it's worth trying.

Answer 2

There are infinitely many systems of ODEs of the form $$\dot x_i(t)=f_i\bigl(t, x_1(t),x_2(t), x_3(t)\bigr)\qquad(1\leq i\leq3),$$ all of them having $\infty^3$ solutions $t\mapsto\phi(t)\in{\mathbb R}^3$, among them the three special $\phi_j$ in your list. The simplest such system is the following:

Your three vector-valued functions $\phi_j$ are solutions of a homogeneous linear system with constant coefficients of the following kind: $$\left[\matrix{\dot x_1\cr\dot x_2\cr\dot x_3\cr}\right] =\left[\matrix{a_{11}&a_{12}&a_{13}\cr a_{21}&a_{22}&a_{23}\cr a_{31}&a_{32}&a_{33}\cr}\right] \left[\matrix{x_1\cr x_2\cr x_3\cr}\right]\ .$$ "Normally" the constant matrix $[A]=[a_{ik}]$ would be given; but here it is unknown, and we have to determine it from the givens. This can be done in the following way: Compute the column vectors $[\dot\phi_j]$. The three vector equations $$[\dot\phi_j(t)]=[A]\,[\phi_j(t)]\qquad(1\leq j\leq3)$$ then expand to nine scalar identities in $t$ involving the matrix elements $a_{ik}$ in a linear way. Comparing coefficients of like terms $e^{-2t}\cos t$, $\>e^{-2t}\sin t$, $\>e^{-3t}$ should then allow you to determine the $a_{ik}$.