Find the general solution by the method of undetermined coefficients.(

1.	Find the general solution by the method of undetermined coefficients.(D3 + 2D2)y = x.
Answer» Factor out as many D’s as possible and write the equation as (D + 2)[D²y] = x. Let z = D²y so the equation is (D + 2)z = x. Since the right-hand side is a polynomial of degree 1, the trial solution should be a polynomial of degree 1, say z = Ax + B. Plugging into the equation gives A + 2Ax + 2B = x. By equating coefficients of powers of x, we get the equations 2A = 1, A + 2B = 0. The solution is A = 1/2, B = −1/4. Thus, we have z = 1/2 x − 1/4. Since z = D²y, we have D²y = 1/2x − 1/4. Integrating once gives Dy = 1/4x 2 − 1/4x and integrating again gives y = 1/12x³ − 1/8x². The solution of the homogeneous equation is yh = C₁ + C₂x + C₃e^2x, so the general solution of the inhomogeneous equation is y = 1/12x³ − 1/8x² + C₁ + C₂x + C₃e^2x,

Find the general solution by the method of undetermined coefficients.(D3 + 2D2)y = x.

Answer»

Factor out as many D’s as possible and write the equation as

(D + 2)[D²y] = x.

Let z = D²y so the equation is

(D + 2)z = x.

Since the right-hand side is a polynomial of degree 1, the trial solution should be a polynomial of degree 1, say z = Ax + B. Plugging into the equation gives

A + 2Ax + 2B = x.

By equating coefficients of powers of x, we get the equations

2A = 1, A + 2B = 0.

The solution is A = 1/2, B = −1/4. Thus, we have

z = 1/2 x − 1/4.

Since z = D²y, we have D²y = 1/2x − 1/4.

Integrating once gives

Dy = 1/4x 2 − 1/4x

and integrating again gives

y = 1/12x³ − 1/8x².

The solution of the homogeneous equation is yh = C₁ + C₂x + C₃e^2x,

so the general solution of the inhomogeneous equation is

y = 1/12x³ − 1/8x² + C₁ + C₂x + C₃e^2x,

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