Solve the equation y y 5 y0 0 y0 0 by using variation of

Solve the equation y” - y’ = 5, y(0) = 0, y(0) = 0, by using variation of parameters.

Solution

given equation y\"-y\'=5

Undetermined coefficients. guessing y = ke^x will give y\" y\'= 0,

so guess y = kxe^x instead:

y = kxe^x

y\'= kxe^x + ke^x

y\" = kxe^x + 2ke^x

y\" y\' = ke^x

k = 1

y = xe^x

let auxiliary equation r²-r=0

r(r-1)=0

r=0 or r=1

therefore the general solution is y=xe^x+C₁+C₂e^x