A differential equation d2y / dx2 - 6 dy / dx + 25y = cos(4x) with y(0) = 0, dy

Question

A differential equation d2y / dx2 - 6 dy / dx + 25y = cos(4x) with y(0) = 0, dy / dx (0) = 0, has a solution of the form y(x) = u(x) + C1v1(x) + C2v2(x), where u(x) is the Particular solution, and C1 and C2 are constants that depend on the given initial condition. What are the solutions to the auxiliary equation? The answer should be one or two numbers separated by commas. The solutions might be complex, e.g. 2+4i,2-4i. v1(x) = Enter an expression to define the function. v2(x) = Enter an expression to define the function. The Particular solution u(x) = Answer is a Expression C1 = This is the coefficient of v1(x) in the general solution. C2 = This is the coefficient of v2(x) in the general solution.

Explanation / Answer

y'' -6y' + 25 y = cos(4x) The auxiliary equation (homogenous solution) is: y'' -6y' + 25 y =0 ===> m^2 - 6m + 25 = 0==> (m^2 -6m +9) +16= 0===> (m-3)^2 =- 4^2 m-3= -4i ==> m= 3-4i AND m-3=4i ==> m= 3+4i yh = A e((3-4i)x) + B e((3+4i)x) yh = A e^(3x) ( cos(4x) - i sin(4x) )+ B e^(3x) ( cos(4x) + i sin(4x) ) yh= e^(3x) ( (A+B) cos(4x) + (Bi - Ai) sin(4x) ) yh= e^(3x) ( C1 cos(4x) + C2 sin(4x) ) v1(x) = e^(3x) cos(4x) AND v2(x) = e^(3x) sin(4x) particular solution : yp= (24 Cos(4 x) Cos(8 x) - 73 Sin(4 x) - 9 Cos(8 x) Sin(4 x) + 9 Cos(4 x) Sin(8 x) + 24 Sin(4 x) Sin(8 x) )/1752 general solution : y(x) = yh + yp = e^(3x) ( C1 cos(4x) + C2 sin(4x) ) + (24 Cos(4 x) Cos(8 x) - 73 Sin(4 x) - 9 Cos(8 x) Sin(4 x) + 9 Cos(4 x) Sin(8 x) + 24 Sin(4 x) Sin(8 x) )/1752 y(x) =( -24 E^(3 x) Cos[4 x] + 24 Cos[4 x] Cos[8 x] - 73 Sin[4 x] + 82 E^(3 x) Sin[4 x] - 9 Cos[8 x] Sin[4 x] + 9 Cos[4 x] Sin[8 x] + 24 Sin[4 x] Sin[8 x]) ) /1752 C1= -24 / 1752 AND C2 = 82 / 1752

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