MATH 20D Lecture 14

Section 9.1 - The Matrix Method

$a{y}^{\prime \prime }(t)+b{y}^{\prime }(t)+cy(t)=f(t),x_1(t)=y(t),x_2(t)=y^{\prime }(t)$

If
From the original ODE:


System we obtain:
Linear:
Non-homogeneous system

$y^{(4)}+y=0,x_1=y,x_2=y^{\prime },x_3=y^{\prime \prime },x_4=y^{\prime \prime \prime }$

From the given ODE:

Linear, homogeneous

$4y^{\prime \prime \prime }+8y^2{y}^{\prime \prime }-\sin (y^{\prime })=0,x_1=y,x_2=y^{\prime },x_3=y^{\prime \prime }$

Non-linear. No normal form

$t^2{y}^{\prime \prime }+(t+1)y^{\prime }+(1-t)y=0,y(1)=2,y^{\prime }(1)=1$

Let and so
From the given ODE:



Linear. Homogeneous
Initial Condition

Transform the following system of first order ODEs into a single equation

Solve for in terms of and in (1).

Then we plug into (2)




Now relabel to get:

Generally, a system of ODEs has the form




If all above are of the form

then the system is called linear. Otherwise, it is a non-linear system.
We can transform linear systems into the form

Here, if then we have a homogeneous system; otherwise, it is non-homogeneous.