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I'm using an integrating factor, rho(x), to solve an equation of the form

dy/dx + P(x)y = Q(x)

I need to find the particular solution.

y' = 1 + x + y + xy; y(0) = 0

y' - y - xy = 1 + x

dy/dx + y(-1-x) = 1 + x

P(x) = (-1-x), Q(x) = (1 + x),

rho(x) = e^(-x-1/2x^2)

(Multiply both sides by rho(x))

e^(-x - 1/2x^2)(dy/dx) + e^(-x - 1/2x^2)(1-x) = e^(-x - 1/2x^2)(1+x)

Dx[y * e^(-x - 1/2x^2)] = e^(-x - 1/2x^2)(1+x)

y * e^(-x - 1/2x^2) = int_e^(-x - 1/2x^2)(1+x)dx

(Multiply both sides by the reciprocal of rho(x))

y(x) = e^(x + 1/2x^2) * int_e^(-x - 1/2x^2)(1+x)dx

(U substitution) This is where I get stuck. It seems that I need u substitution to find the integral on the right side

u = e^(-x - 1/2x^2)

du = (-x - 1) * e^(-x - 1/2x^2)

dv = 1

v = (1+x)

When I set that up in uv - int_v du it looks just as bad as the original integral.

Thanks in advance for any assistance.

dy/dx + P(x)y = Q(x)

I need to find the particular solution.

y' = 1 + x + y + xy; y(0) = 0

y' - y - xy = 1 + x

dy/dx + y(-1-x) = 1 + x

P(x) = (-1-x), Q(x) = (1 + x),

rho(x) = e^(-x-1/2x^2)

(Multiply both sides by rho(x))

e^(-x - 1/2x^2)(dy/dx) + e^(-x - 1/2x^2)(1-x) = e^(-x - 1/2x^2)(1+x)

Dx[y * e^(-x - 1/2x^2)] = e^(-x - 1/2x^2)(1+x)

y * e^(-x - 1/2x^2) = int_e^(-x - 1/2x^2)(1+x)dx

(Multiply both sides by the reciprocal of rho(x))

y(x) = e^(x + 1/2x^2) * int_e^(-x - 1/2x^2)(1+x)dx

(U substitution) This is where I get stuck. It seems that I need u substitution to find the integral on the right side

u = e^(-x - 1/2x^2)

du = (-x - 1) * e^(-x - 1/2x^2)

dv = 1

v = (1+x)

When I set that up in uv - int_v du it looks just as bad as the original integral.

Thanks in advance for any assistance.

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