Question

X^2y''-2y=2x-1 find the solution of nonhomogeneous equations

Answer 1

$+ - = -$

Answer 2

Answer: The solution of the differential equation is y = Cx²+ D/x - x + 1/2

Step-by-step explanation:

Step1: Finding homogeneous solution

x²y" - 2y = 0

Substitute $y = x^m$

$x^m$ [m(m-1) - 2] = 0

m² - m - 2 = 0

(m-2)(m+1) = 0

m = 2 , -1

Yh = Cx²+ D/x

Step 2: Finding particular solution

x2y" -2y = 2x-1

Substitute y = Ax+B

0 - 2Ax -2B = 2x -1

-2Ax -2B = 2x -1

Comparing the coefficients on both sides,

-2A = 2

A= -1

-2B = -1

B = 1/2

Yp = -x + 1/2

Step 3: Writing general solution

y = Yh + Yp

y = Cx²+ D/x - x + 1/2

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