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Just learned that if $y_1$ and $y_2$ are solutions to a homogeneous equation, then so a linear combination of $y_1$ and $y_2$. Now, I am sure, but don't know enough if there is some geometric reasoning.

The reason that I am asking is because for the solutions for the constants $c_1$ and $c_2$ uses matrix.

$y(t_0)=y_0, \quad y^{\prime}(t_0)=y^{\prime}_0$

and $c_1$ and $c_2$ satisfy the equations

$$ \begin{aligned} c_1y_1(t_0)+c_2y_2(t_0)&=y_0 \\[0.4em] c_1y^{\prime}_1(t_0)+c_2y^{\prime}_2(t_0)&=y^{\prime}_0 \end{aligned} \qquad \implies \qquad \begin{aligned} c_1=\frac{\begin{vmatrix} y_0 & y_2(t_0)\\ y^{\prime}_0 & y^{\prime}_2(t_0) \end{vmatrix}}{ \begin{vmatrix} y_1(t_0) & y_2(t_0)\\ y^{\prime}_1(t_0) & y^{\prime}_2(t_0)\end{vmatrix} } \\[0.4em] c_2=\frac{\begin{vmatrix} y_1(t_0) & y_0\\ y^{\prime}_1(t_0) & y^{\prime}_0 \end{vmatrix}}{ \begin{vmatrix} y_1(t_0) & y_2(t_0)\\ y^{\prime}_1(t_0) & y^{\prime}_2(t_0) \end{vmatrix}} \end{aligned} $$

I am thinking that there has to be a relation between vector spaces and these family of solutions, but not sure.

Am I incorrect?

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Yes, you are correct, the family of solutions of a homogenous linear differential equation forms a vector space, a subspace of the space of twice continuously differentiable functions. The set of solutions of an inhomgenous ODE is then an affine subspace, a shifted version of the homogenous solution set.

The Wronski matrix and their properties tells you that any system of solutions and their linear dependence or independence is completely determined by the dependence pattern in one point. So the solutions whose initial conditions form the columns of the identity matrix are already a basis of the solution space and any solution is a linear combination of them.

The same holds of course for any system of independent solutions, where you then have to solve a system of linear equations to translate the initial conditions into coefficients in that basis. But using Cramers rule is the most inefficient way to do that if the order of the differential equation becomes larger than $3$.

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