User:Egm4313.s12.team13.Neisha.R2

Problem R 2.5B
'''pg.59 #17

Problem Statement:

17. Find an ODE for the given solutions.

$$ e ^{- \sqrt{5} x} $$, $$ e ^{- \sqrt{5} x} $$

using $$ \; {y}''+a {y}'+b y=0 $$ for the given basis.

Solution:

Case: Real double root ODE

where $$ \; \lambda = -a/2 $$

and the general solution is $$ \; y= e ^ {-ax} (c_1+ c_2x)  $$

So $$   \; e ^ { \lambda x} =  e ^ {- \sqrt {5} x}  $$

$$  \; e ^ {-ax/2} =  e ^ {- \sqrt {5} x}  $$

Solving for a,

$$  \;  -ax/2= - \sqrt {5} x  $$

$$  \; a= 2 \sqrt {5}  $$

By plugging into the determinant $$ \; a^2-4b=0 $$, Solve for b.

$$  \; {2 \sqrt {5}} -4b=0  $$

$$  \; 4(5)=4b $$

$$  \; b=5  $$

By plugging in a and b, the ODE is:

$$  \; {y}''+ 2 \sqrt {5}{y}'+ 5y=0 $$ 