Find the general solution of the given differential equation

Find the general solution of the given differential equation. (x+1)dy/dx+(x+2)y = 2xe^-x y= Give the largest interval I over which the general solution is defined. (Think about the implications of any singular points.) Determine whether there are any transient terms in the general solution. (Enter the transient terms as a comma-separated list; if there are none, enter NONE.)

Solution

Divide all over by (x+1) :

dy/dx + (x+2)/(x+1) * y = 2xe^-x / (x + 1)

IF = e^(integral of (x+2)/(x+1)*dx)

IF = e^(int of (1 + 1/(x+1))*dx)

IF = e^(x + ln|x + 1|)

IF = (x + 1)e^(x)

Multiply all over by the Integrating factor :

(x + 1)e^(x) * [dy/dx + (x+2)/(x+1) * y = 2xe^-x / (x + 1)]

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

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

Integrating :

y * (x + 1)e^(x) = x^2 + C

y = (x^2 + C) / ((x + 1)e^x) ---> SOLUTION

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Clearly, since we have the x + 1 in dneominator, x cannot be equal to -1....
This splits number line into intervals (-inf , -1) and (-1 , inf)
out of which (-1 , inf) is LARGER

So, (-1 , inf) ---> ANSWER

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Both terms, x^2 / ((x + 1)e^x) and C / ((x + 1)e^x) are transient because as x ---> infinity, both of these terms tends to 0

x^2 / ((x + 1)e^x) , C / ((x + 1)e^x) ---- --> ANSWER