Answer:
[tex]y(x)=c_1e^{-\frac{5}{2}x}+c_2xe^{-\frac{5}{2}x}[/tex]
Step-by-step explanation:
The given differential equation is 4y"+20y'+25y = 0
The characteristics equation is given by
[tex]4r^2+20r+25=0[/tex]
Now, solve the equation for r
Factor by middle term splitting
[tex]4r^2+10r+10r+25=0\\\\2r(2r+5)+5(2r+5)=0[/tex]
Factored out the common term
[tex](2r+5)(2r+5)=0[/tex]
Use Zero product property
[tex](2r+5)=0,(2r+5)=0[/tex]
Solve for r
[tex]r_{1,2}=-\frac{5}{2}[/tex]
We got the repeated roots.
Hence, the general equation for the differential equation is
[tex]y(x)=c_1e^{-\frac{5}{2}x}+c_2xe^{-\frac{5}{2}x}[/tex]
