DIFFERENTIAL EQUATIONS PROBLEM Solve the ODE Solve the ordin

DIFFERENTIAL EQUATIONS PROBLEM

Solve the ODE>

Solve the ordinary differential equation y;\' = y = e^x (x cos x - (3 + 2x) sin x)

Solution

Finding the solution of the differential equation : y\'\' + y = 0

For general solution

Let , y = e^rt

=> ( r² + 1 )e^rt = 0

=> r = i or -i , where i = square root of unity

=> y = c₁ cos(x) + c₂ sin(x)

For particular solution , assume that y_p = a.e^x sin(x) + bx.e^x cos(x)

=> y_p\'\' = 2a.e^x.cos(x) + 2b.e^x.( cos(x) - (x+1)sin(x) )

=> y_p\'\' + y_p = e^x ( cos(x)*( 2a + 2b ) + sin(x)*( - 2bx - 2b + a ) )

Comparing with the original equation

=> b = 1 , a = -1

=> y = c₁ cos(x) + c₂ sin(x) - e^x sin(x) + x.e^x cos(x)