Find the General Solution to the ODE 2y5y3y0Solution2y5y3y0

Find the General Solution to the ODE:

2y\'\'\'-5y\'\'-3y\'=0

Solution

2y\'\'\'-5y\'\'-3y\'=0

[2D³-5D²-3D] y = 0

The auxiliary equation is 2m³-5m²-3m = 0

                                 m [ 2m²-5m-3] = 0

                        m =0      2m²-6m+m-3 = 0

2m(m-3)+ (m-3) = 0

   (m-3) (2m+1) = 0

   m-3 = 0    2m+1 = 0

                                     m = 3, m = -1/2

        Therefore,

              m =0, -1/2, 3

Hence, the general solution is

            y = c₁e^0x + c₂ e^(-1/2)x + c₃e^3x

       i.e. y = c₁+ c₂ e^(-1/2)x + c₃e^3x