Question

solve d^2y/dx^2 -5dy/dx +6y = e^4x​

Answer 1

Answer:

x2d2ydx2−4xdydx+6y=x4x2d2ydx2−4xdydx+6y=x4

This is a Cauchy-Euler differential equation. This means the complementary solution ycyc has fundamental solutions yk=xryk=xr.

Differential Equations - Euler Equations

Compute the derivatives and substitute it into the homogenous differential equation.

yk=xr,y′k=rxr−1,y′′k=r(r−1)xr−2yk=xr,yk′=rxr−1,yk″=r(r−1)xr−2

x2d2ydx2−4xdydx+6y=0x2d2ydx2−4xdydx+6y=0

x2r(r−1)xr−2−4xrxr−1+6xr=0x2r(r−1)xr−2−4xrxr−1+6xr=0

Notice how this choice of yk=xryk=xr

Step-by-step explanation:

This is a Cauchy-Euler differential equation. This means the complementary solution ycyc has fundamental solutions yk=xr

Ans:- zero