solve it by integating factor
dy/dx + (x+1/x)y=e^-2x
To solve the given differential equation using an integrating factor, you will need to follow these steps:
1. Identify the differential equation in the standard form: \(dy/dx + P(x)y = Q(x)\), where \(P(x)\) and \(Q(x)\) are functions of \(x\).
In our case, the differential equation is \(dy/dx + (x+1)/x)y = e^{-2x}\), with \(P(x) = (x+1)/x\) and \(Q(x) = e^{-2x}\).
2. Compute the integrating factor \(I(x)\) using the formula: \(I(x) = e^{\int P(x) dx}\).
In this case, \(P(x) = (x+1)/x\), so we calculate the integral of \(P(x)\) as follows:
\(\int P(x) dx = \int \frac{x+1}{x} dx\)
Performing the integration gives us:
\(\int \frac{x+1}{x} dx = \int \left(1 + \frac{1}{x}\right) dx = x + \ln|x| + C\),
where \(C\) is the constant of integration.
Therefore, the integrating factor is \(I(x) = e^{x + \ln|x| + C} = e^C \cdot e^{x + \ln|x|} = Ce^x |x|\), where \(C = e^C\).
3. Multiply both sides of the differential equation by the integrating factor \(I(x)\).
Multiply the given equation \(dy/dx + (x+1)/x)y = e^{-2x}\) by \(Ce^x |x|\):
\(Ce^x |x|(dy/dx) + C(x+1)y = Ce^x |x| e^{-2x}\).
Simplifying, we get:
\((Ce^x |x|y)' = Ce^{-x} |x|\).
4. Integrate both sides of the equation with respect to \(x\).
Integrating both sides yields:
\(\int (Ce^x |x|y)' dx = \int Ce^{-x} |x| dx\).
Using the fundamental theorem of calculus, the left side becomes:
\(Ce^x |x|y = -Ce^{-x} |x| + C_1\),
where \(C_1\) is the constant of integration.
5. Solve for \(y\).
Divide both sides of the equation by \(Ce^x |x|\):
\(y = \frac{-Ce^{-x} |x| + C_1}{Ce^x |x|}\).
Simplifying further, we get:
\(y = \frac{-e^{-x} |x| + C_1}{e^x |x|}\),
or equivalently,
\(y = \frac{-1}{x} + \frac{C_1}{x e^x}\).
Therefore, the solution to the given differential equation is \(y = \frac{-1}{x} + \frac{C_1}{x e^x}\), where \(C_1\) is the constant of integration.