The linear first order differential equation:
"\\frac {dy}{dx} + P(x)y = Q(x)"
has the integrating factor (IF) : "IF= e^{\\int P (x) dx}" .
(1) "\\frac {dy}{dx}+y=e^{x}" .
Integrating factor:
"P(x) = 1" .
Integrating factor, "IF= \\int e^{\\int P (x) dx} = e^{\\int dx}= e^{x}"
"IF= e^{x}"
Multiply equation by IF:
"e^x\\frac {dy}{dx} + e^xy = e^xe^x"
So,
"e^x\\frac {dy}{dx} + e^xy = e^{2x}" ,
i.e. "\\frac {d}{dx}[ e^xy] = e^{2x}"
Integrate:
"e^xy = \\int e^{2x} dx"
i.e.
"e^xy = \\frac {1}{2}e^{2x} + C ,"
.
Therefore, general solution of a Differential Equation (1):
.
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