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Solve the given problem by the method of variation of parameters, y'' + 2y' + y = e^(-x)lnx.

User Spanky
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Answer:

The method of variation of parameters is used to solve second-order linear differential equations with non-homogeneous terms. The given differential equation is:

y'' + 2y' + y = e^(-x)ln(x)

First, let's find the complementary solution (homogeneous solution) for the associated homogeneous equation, which is:

y'' + 2y' + y = 0

The characteristic equation is:

r^2 + 2r + 1 = 0

This equation has a repeated root of -1:

(r + 1)(r + 1) = 0

So, the complementary solution is:

y_c(x) = (A + Bx)e^(-x)

Now, we need to find the particular solution (y_p) for the non-homogeneous equation. You can use the method of variation of parameters for this. Assume the particular solution is of the form:

y_p(x) = u(x)(A + Bx)e^(-x)

Now, find y_p'' and y_p':

y_p'' = u''(A + Bx)e^(-x) + 2u'(A + Bx)e^(-x) + u(A + Bx)e^(-x)

y_p' = u'(A + Bx)e^(-x) + u(A + Bx)e^(-x)

Now, substitute y_p, y_p', and y_p'' into the original equation:

[u''(A + Bx)e^(-x) + 2u'(A + Bx)e^(-x) + u(A + Bx)e^(-x)] + 2[u'(A + Bx)e^(-x) + u(A + Bx)e^(-x)] + [u(A + Bx)e^(-x)] = e^(-x)ln(x)

Now, simplify and collect terms with the same coefficients of u, u', and u'':

u''(A + Bx)e^(-x) + 4u'(A + Bx)e^(-x) + 3u(A + Bx)e^(-x) = e^(-x)ln(x)

Now, you need to choose u(x) so that the term with u'' simplifies to e^(-x)ln(x). Let's choose:

u''(A + Bx)e^(-x) = e^(-x)ln(x)

So, we have:

u'' = ln(x)

Integrate u'' to find u':

u' = ∫ln(x) dx

Integrating ln(x), we get:

u' = xln(x) - ∫x(1/x) dx

u' = xln(x) - ∫dx

u' = xln(x) - x + C

Now, integrate u' to find u:

u = ∫(xln(x) - x + C) dx

u = ∫xln(x) dx - ∫x dx + ∫C dx

Integrating xln(x) and x, we get:

u = (1/2)x^2ln(x) - (1/2)∫x^2 dx + Cx

Now, you have u(x), and you can find y_p:

y_p(x) = u(x)(A + Bx)e^(-x)

y_p(x) = [(1/2)x^2ln(x) - (1/2)∫x^2 dx + Cx](A + Bx)e^(-x)

Now, you can simplify and find the constants A, B, and C to match initial conditions or boundary conditions if provided. This will give you the complete solution to the differential equation.

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User Amr Ellafy
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