solve the equation: (x^2 + 3x + 2)dy/dx + xy = x(x + 1)
dy/dx + x/(x^2 + 3x + 2)*y = x(x+1)/(x^2 + 3x + 2)
dy/dx + x/{(x + 2)(x + 1)}*y = x(x+1)/{(x + 2)(x + 1)}
dy/dx + x/{(x + 2)(x + 1)}*y = x/(x + 2)
I.F. = e^{int x/{(x + 2)(x + 1)} dx}
Decomposing the integrand into separate fractions, we get,
I.F. = e^{-ln(x+1)+2ln(x+2)} = e^{ln(x+2)^2 – ln(x+1)}
I.F. = (x+2)^2/(x+1)
Using this I.F., the DE can now be written as,
d((x+2)^2/(x+1)*y)/dx = x/(x + 2) *(x+2)^2/(x+1) = x(x+2)/(x+1)
(x+2)^2/(x+1)*y = int x(x+2)/(x+1) dx
Again, decomposing the integrand into separate fractions, we get,
(x+2)^2/(x+1)*y = ½*x^2 + x – ln(x+1) +const
y(x) = (x+1)/(x+2)^2*{1/2*x(x+2) – ln(x+1) + C}
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