Dofferential Equation

Nov 2009
147
21
Find the solution of the differential equation,

\(\displaystyle \frac{d^4y}{dx^4}-4 \cdot \frac{d^3y}{dx^3}+8 \cdot \frac{d^2y}{dx^2}-8 \cdot \frac{dy}{dx}+4y=0\)

Pl help to solve it. Thanks in adv..
 
May 2009
959
362
The characteristic equation is \(\displaystyle r^{4}-4r^{3}+8r^{2}-8r+4 = (r^{2}-2r+2)^{2} = 0 \)

which has roots \(\displaystyle 1 \pm i \), both of multiplicity 2

so the general solution is \(\displaystyle y(x) = C_{1}e^{x} \cos(x) + C_{2} e^{x} \sin x + C_{3} e^{x}x \cos x + C_{4} e^{x}x \sin x \)
 
  • Like
Reactions: kjchauhan