Solve each of the following differential equations y9y 14 y

Solve each of the following differential equations: y\"-9y +14 y- 4e^2x.

Solution

y\'\'-9y\'+14y=4e^2x ................(*)

we know the general solution of the given differential equation is-

y=y_h+y_p

to find y_h,we write the characteristic equation and finding the roots, we get

m²-9m+14=0

m= 2,7

hence,

y_h=c₁e^2x+c₂e^7x

now to find the particular solution y_p,

as we already have e^2x in our homogeneous solution..so we take

y_p=Axe^2x

y_p\'=Ae^2x+2Axe^2x

y_p\'\'=2Ae^2x+2Ae^2x+4Axe^2x = 4Ae^2x+4Axe^2x

putting these values in equation (*),we get

(4Ae^2x+4Axe^2x)-9(Ae^2x+2Axe^2x)+14Axe^2x=4e^2x

4Ae^2x+4Axe^2x-9Ae^2x-18Axe^2x+14Axe^2x=4e^2x

-5Ae^2x=4e^2x

A=-4/5

hence,

y_p=-4xe^2x/5

so,the general solution is--

y=c₁e^2x+c₂e^7x-4xe^2x/5