# Second Order Linear Differential Equations

1). Given the differential equation for

1. L[y]= y''+2by'+b2y = exp(-bx)/x2, x>0 ;

a) Find the complementary solution of (1) by solving L[y] = 0.

b) Solve (1) by introducing the transformation y[x]= exp(-bx) v(x).

into (1) and obtaining and solving completely a differential equation for v(x) . Now identify the particular solution of (1).

2). Using the Method of Undetermined coefficients find a particular solution of y''+y = x2 + sin(x)

3). Determine the functional dependence in x of the particular solutions of the following differential equations. Do Not solve for the constants.

a) y''-5y'+6y = exp(x)cos(2x)+exp(2x)*(3x+4)*sin(x)

b) y''+2y'+2y = 5exp(-x)*x2*cos(x) + 4exp(-x)*sin(3x)

c) y''+4y = 3cos(x) + x sin(2x)

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Three differential equations problems are solved in complete step-by-step detail. Trial functions for particular solutions are covered in greatest detail. Solution is given in both Word .doc format (with Mathtype 4 equations) or in pdf format.

General solution of non-homogenous second order linear differential equations : d²y/dx² + y = cosx

Please find the general solution of the nonhomogenous second order linear differential equation below by following these steps:

1. Find the general solution y= C1y1 + C2y2 of the associated homogenous equation (complementary solution)

2. Find a single solution of yp of above.(particular solution).

3. Express the general solution in the form y=yp + C1y1 +C2y2

Please show all steps. Thank you.

d²y/dx² + y = cosx

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