Find the general solution of the given differential equation

Find the general solution of the given differential equation. X^2y + x(x + 2)y = eX Give the largest interval I over which the general solution is defined. (Think about the implications of any singular points.) (1, Infinity) (-1, 1) (-Infinity, Infinity) (0, Infinity) (0, 1) 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^2 :

y\' + ((x+2)/x)*y = e^x / x^2

y\' + (1 + 2/x)*y = e^x / x^2

Integrating factor = e^(integral of (1 + 2/x))

e^(x + 2ln(x))
e^x * e^(2ln(x))
e^x * e^ln(x^2)
x^2*e^(x)

Multiply all over by the IF :

x^2e^x * [y\' + (1 + 2/x)*y = e^x / x^2]

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

Integrating :

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

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

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Clearly, x cannot equal 0 because if so, it would result in 0 in denominator, which is impossible in math...

So, (0 ,infinity) ---> ANSWER

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Yes, we do have transient terms, i.e the C/(x^2*e^x) term....

IT is so because as x becomes too large, x^2 * e^x becomes extremely large

And thus C / x^2*e^x becomes extremely snall, i.e almost ZERO