Find the general solution to the differential equation y5y7y

Find the general solution to the differential equation.

y\'\'\'-5y\'\'+7y\'-3y

Solution

we have a linear homogeneous ode with constant coefficients so we assume solution of the form

y=exp(kx)

Substituting gives us the characteristic equation

k^3-5k^2+7k-3=0

Setting k=1 we see k=1 is a root (1-5+7-3=0)

Usign this informatio we can rewrite the equation as

k^3-k^2-4k^2+4k+3k-3=0

k^2(k-1)-4k(k-1)+3(k-1)=0

(k-1)(k^2-4k+3)=0

k=1,3

k=1 is a repeated root

So genearal solution is

y=e^x(A+Bx)+e^{3x}