Question

y" + 2y' = 2x + 5 - e-2x

Answer 1

THE GENERAL DOLUTION OF THE DIFFERENTIAL EQUATION::::::::::::

Solve dx The general solution will be the sum of the complementary solution and particular solution Find the complementary solution by solving Assume a solution will be proportional to Substitute y(x) e1x into the differential equation +2 42 C1x for some constant λ. dx S ubstitute d-2 (car) sex and dre")-λ c1x Factor out c* Since e1x * 0 for any finite λ, the zeros must come from the polynomial Factor: A(λ + 2) 0 Solve for λ: The root λ =-2 gives y1(x) = C1 e-2x as a solution, where ci is an arbitrary constant The root A 0 gives 2(x)2 as a solution, where c2 is an arbitrary constant.

The general solution is the sum of the above solutions: Determine the particular solution to dry, + 2 by the method of undetermined coefficients: =-e-2x + 2x + 5 The particular solutionwill be the sum of the particular solutions to +24 dx The particular solution to d2yo +2 dx2x5i of the form: dx yP (x)x(a1 +a2 x), where a1 +a2 x was multiplied by X to account for 1 in the complementary solution The particular solution to Yp2 (x) -х (аз e-2x), where@g С-2x was multiplied by +2x-e2x is of the form: dx x to account for e-2in the complementary solution. Sum yp (x) ndyp2(x) to obtain yp(x) Solve for the unknown constants a, a, and a3 Compute -2 x -2 x d2yp) Compute d

Substitute the particular solution ypx) into the differential equation 2 x -e- + 2x + 5 Simplify: (2a1+2a2)-2a3c-2 x + 4a2 x = 5-c-2 x + 2x Equate the coefficients of 1 on both sides of the equation 2a1 + 2a2=5 Equate the coefficients of e2* on both sides of the equation Equate the coefficients of x on both sides of the equation 4a2 2 Solve the system Substitute a 1.a2, and a3 into yp(x-a, x + a2 x2 + аз e-2xx: yp(x) + 2x + e-2xx 2 2

The general solution is: y(x) ye(x) + yp(x)-xf + 2 x + 1 e-2x x + c 1 e-2x + c2