Linear structure of linear ODEs in ordinary language

A linear differential equation of order $n$ is as follows: $$\begin{split} &(\textrm{some function of }t) \cdot y^{(n)} + \cdots + (\textrm{(another function of }t) \cdot y '' + \\ &(\textrm{yet another function of }t) \cdot y'+ (\textrm{once more a function of }t) \cdot y + \\ &(\textrm{and guess what, another function of }t) = 0 \end{split}$$ This equation is called homogeneous if there is no term that depends only on $t$ (and not of $y$ or one of its derivatives).

Now you want to find solutions to the non-homogeneous differential equation above. Suppose you (accidentally or by an educational guess) have found one solution $y_{\text{part}}$ (called a particular or specific solution of the ODE). Moreover, suppose that you have solved the homogeneous equation (so you deleted the term that depended only on $t$ and solved that equation); call this solution $u$.

Then EVERY solution of the non-homogeneous differential equation can be written as $y_{\text{part}}+u$. This helps tremendously! It means that you know the complete solution of a non-homogeneous ODE once you know one particular solution and know the general solution of the corresponding homogeneous differential equation

For the homogeneous differential equation, with no term that depends only on $t$, the following is true:

• the constant function $u(t)=0$ is a solution;
• if $u(t)$ and $v(t)$ are two solutions of the homogeneous ODE, then any linear combination of these two solutions is also a solution of the ODE. Thus, for any numbers $\alpha$ and $\beta$, and for any solutions $u$ and $v$ of the homogeneous ODE, the linear combination $\alpha\cdot u+\beta\cdot v$ is a solution, too.

Linear structure of linear ODEs in mathematical language

Let $a_0(t), a_1(t),\ldots ,a_n(t)$ and $b(t)$ be continuous functions with $a_n(t)\ne0$ such that $$a_n(t)\cdot y^{(n)}+\cdots+a_2(t)\cdot y''+a_1(t)\cdot y'+a_0(t)\cdot y +b(t)=0$$ is a linear differential equation of order $n$ is. This ODE is homogeneous if and only if $b(t)=0\tiny.$

Suppose that $y_{\text{part}}$ is a solution of the non-homogeneous differential equation. Then any other solution can be written as $y_{\text{part}}+u$, where $u$ is a solution of the corresponding homogeneous ODE, i.e., of the ODE with $b(t)$ replaced by $0$.

We call $y_{\text{part}}$ a particular solution or a specific solution of the ODE.

If $b(t)=0$, then the set of solutions is linear in the following sense:

• the constant function $u(t)=0$ is a solution;
• if $\alpha$ and $\beta$ are real numbers, and if $u$ and $v$ are solutions of the corresponding homogeneous ODE, then $\alpha\cdot u+\beta\cdot v$ is a solution, too.