### Differential equations: Introduction to Differential equations

### Linear ODEs

We will now pay attention the following type of differential equation.

Let #\varphi(t,y,y',y'',\ldots)=0# be an ordinary differential equation in the unknown function \(y\) of the single variable \(t\).

- The differential equation is called
**linear**if the function \(\varphi\) can be written as a sum of terms that are either a function of #t# or a product of a function of #t# with \(y\) or a (higher) derivative of #y#. Otherwise, the ODE is called**nonlinear**. - A linear differential equation is called
**homogeneous**if each term in the equation is a multiple of #y# or of a derivative of #y#. Otherwise the linear ODE is called**non-homogeneous**or**inhomogeneous.**

A linear differential equation of order #3# is of the form

\[a_3(t)\cdot y'''+a_2(t)\cdot y''+a_1(t)\cdot y'+a_0(t)\cdot y +b(t)=0\]

where #a_3#, #a_2#, #a_1#, #a_0#, and #b# are functions. The ODE is homogeneous if and only if #b=0#.

The following statement explains the term "linear" for an ODE and helps solving them.

Linear structure of linear ODEs

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#. This equation is homogeneous if and only if #b(t)=0#.

Suppose that #y_{\text{part}}# is a solution of this equation. Then every other solution has the form

\[u+y_{\text{part}}\] where #u# is a solution of the corresponding homogeneous ODE (that is, the equation with #b(t)# replaced by #0#).

We call the solution #y_{\text{part}}# a **particular solution** of the ODE.

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

- The constant function #u=0# is a solution.
- If #\alpha# and #\beta# real numbers and #u# and #v# are solutions of the homogeneous ODE, then so is #\alpha\cdot u+\beta\cdot v#.

If the initial ODE is homogeneous, then we take #y_{\text{part}}=0# to be the particular solution.

In order to find a particular solution #y_{\text{part}}# of a linear ODE, we can make additional assumptions that simplify the search. We must then assess the compliance of the solutions to those assumptions. Below are some examples. For the solution of the corresponding homogeneous equation, at least in the case of order #1# and in special cases for order #2#, we will present a general method.

We replace #y'# by #u# and #y''# by #v# in the left side of the equation

\[ y\cdot \lambda+v\cdot {\it x }+u\cdot \left(-x+\mu+1\right) \]

This is a linear expression in the variables #y#, #u#, and #v# . Therefore, the ODE is linear.

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