Question

Find the general solution of the given second-order differential equation. y'' + 10y' + 25y = 0

Solve the given differential equation by undetermined coefficients. y'' + 4y = 2 sin 2x

Solve the given differential equation by undetermined coefficients. y'' − y' = −10

Answer 1

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Consider the differential equation (DE) y" +4y = 2 sin 2x . The auxillary equation is m+4 0 m = +2i Thus, the complementary solution is ,弌1 cos 2x + c, sin 2x To find the particular solution, let Then y,- A cos 2x-2Ar sin 2x Bsin 2r +2Br cos 2.x y2A sin 2x2B cos 2x 2A sin 2x - 4Ax cos 2.x +2B cos 2.x -4Bx sin 2x =-4A sin 2x + 4B cos 2x-4.4x cos 2x-4Br sin 2x Substituting these values in yz +4y, -2sin 2.x gives 2 Sin 2x -4A sin 2x + 4B cos 2x-4.4x cos 2x-4B.x sin 2x + 4 ( Ax cos 2x + Br sin 2x ) 2 sin 2x -4.4 sin 2x + 4B cos 2x = 2 sin 2x Equate the coefficients of like terms on both sides Solving the equations gives A =--B = 0 2 Thus, the particular solution is y,xcos 2.x. 2 Therefore, the general solution to the IVP is y =y, +y, = | c, cos 2x + c, sin 2x- x cos 2x 2

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