Differential equations and Cramer's rule

SilverKing

New member
Joined
Dec 25, 2013
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23
Hi everyone,

I'm taking the Differential Equations for the first time, and I want to know the most helpful textbook for the subject.

We had the following example:
Find the differential equation which its general solution is:
y=C1+C2x+x^2

Solution:
y'=0+Cx+2x
y''=0+0+2

Solving:

\begin{bmatrix}
y-x^{2} & 1 & x \\
y^{'}-2x & 0 & 1 \\
y^{''}-2 & 0 & 1
\end{bmatrix}

y''=2


How and why did the equations become like that?
 
If you are dealing with differential equations, I presume you know that the derivative of \(\displaystyle y= C_1+ C_2x+ x^2\) is \(\displaystyle y'= C_2+ 2x\). That is not yet a general equation because it still involves the constant \(\displaystyle C_1\). But differentiating again, \(\displaystyle y''= 2\). That now has no undetermined constants so is the equation that will give the original polynomial as solution.

I have no idea where you got that matrix or what it has to do with this equation.
 
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