# Exactly one homogeneous differential equation of second order to given fundamental solution

I am working on:

Let $$\phi_1,\phi_2$$, so that $$\phi_1(x)\phi_2'(x)-\phi_1'(x)\phi_2(x)\neq 0.$$ for all $$x\in\mathbb{R}$$. Then there exists exactly on homogeneous differential equation of second order $$y''(x)=f(x)y'(x)+g(x)y(x)$$ so that the functions $$\phi_1,\phi_2$$ are fundamental solutions.

My idea is to suppose that there are two homogeneous ODE of second order. Is there a possibility to combine the terms, so I can use $$W(x)\neq 0$$?

I think your idea works to prove uniqueness. Suppose there exist functions $$f_1, f_2, g_1, g_2$$ such that $$\phi_1$$ and $$\phi_2$$ are fundamental solutions to the second order, homogeneous ordinary differential equations $$y'' + f_1(x)y' + g_1(x)y = 0$$ and $$y'' + f_2(x)y' + g_2(x)y = 0$$. This means that the general solution of these differential equations are of the form $$y(x) = c_1\phi_1(x) + c_2\phi_2(x)$$ for constants $$c_1, c_2$$. In particular, $$y(x) = \phi_1(x)$$ and $$y(x) = \phi_2(x)$$ are solutions to these differential equations. So $$\phi_1'' + f_1(x)\phi_1' + g_1(x)\phi_1 = 0 \quad \text{and} \quad \phi_2'' + f_1(x)\phi_2' + g_1(x)\phi_2 = 0,$$ and $$\phi_1'' + f_2(x)\phi_1' + g_2(x)\phi_1 = 0 \quad \text{and} \quad \phi_2'' + f_2(x)\phi_2' + g_2(x)\phi_2 = 0.$$ Subtracting, you get $$(f_1(x) - f_2(x))\phi_1' + (g_1(x) - g_2(x))\phi_1 = 0 \quad \text{and} \quad (f_1(x) - f_2(x))\phi_2' + (g_1(x) - g_2(x))\phi_2 = 0.$$ You can the rewrite this as a matrix equation: $$\begin{bmatrix} \phi_1'(x) & \phi_1(x) \\ \phi_2'(x) & \phi_2(x) \end{bmatrix}\begin{bmatrix} f_1(x) - f_2(x) \\ g_1(x) - g_2(x) \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}.$$ For any fixed $$x \in \mathbb{R}$$, the $$2 \times 2$$ matrix is invertible as your Wronskian is non-zero. This means that for any $$x \in \mathbb{R}$$, $$f_1(x) = f_2(x)$$ and $$g_1(x) = g_2(x)$$, which shows that if there exists a second order homogeneous ordinary differential equation for which $$\phi_1$$ and $$\phi_2$$ are fundamental solutions, then it must be unique.