Find the general solution of: y'' - 4y' + 4y = e^(2x)/x² using variation of parameters.
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Step-by-Step Solution
Step 1: Find the complementary solution
First, we find the complementary solution yc by solving the homogeneous equation y′′−4y′+4y=0. We form the characteristic equation m2−4m+4=0, which factors as (m−2)2=0. This gives us a repeated root m=2. Therefore, the complementary solution is yc=c1e2x+c2xe2x.
Step 2: Identify y1, y2 and f(x)
From the complementary solution, we identify y1=e2x and y2=xe2x. The non-homogeneous term f(x) is e2x/x2.
Step 3: Calculate the Wronskian
Next, we calculate the Wronskian W(y1,y2). The Wronskian is the determinant of the matrix formed by y1, y2 and their first derivatives. After computing the determinant, we find that the Wronskian is e4x.
Step 4: Calculate u1′ and u2′
Using the formulas for variation of parameters, we calculate u1′ and u2′. We substitute y1, y2, f(x), and W into the formulas. After simplification, we get u1′=−1/x and u2′=1/x2.
Step 5: Integrate to find u1 and u2
Now, we integrate u1′ and u2′ to find u1 and u2. The integral of −1/x is −ln∣x∣, and the integral of 1/x2 is −1/x. We omit the constants of integration for the particular solution.
Step 6: Form the particular solution yp
The particular solution yp is given by u1y1+u2y2. Substituting the expressions for u1, u2, y1, and y2, we get yp=−e2xln∣x∣−e2x.
Step 7: Write the general solution
Finally, the general solution is the sum of the complementary solution and the particular solution, y=yc+yp. We combine the results to get the final general solution.