dy/dx+((2x+1)/x)y=e^-2x
This is a good example a first-order linear d.e.
y' + P(x)y = Q(x)
where
P(x) = (2x+1)/x
Q(x) = e^-2x
Int(2x+1)/x dx = Int(2 + 1/x dx) = 2x + lnx
e^(2x+lnx) = xe^2x
so an integrating factor is xe^2x, giving us
xe^2x*y' + xe^2x * (2x+1)/x y = xe^2x * e^-2x
d/dx(xe^2x y) = x
so,
xe^2x y = x^2/2 + C
y = e^-2x (x/2 + C/x)
Great! Thank you I understand now :)
This is a first-order linear ordinary differential equation. To solve it, we can use an integrating factor. The integrating factor is found by multiplying the equation by an appropriate term to make the left side of the equation a total derivative.
First, let's rearrange the equation in standard form:
dy/dx + ((2x + 1)/x)y = e^(-2x)
Now, we compare this to the standard form:
dy/dx + p(x)y = q(x)
In this case, p(x) = (2x + 1)/x and q(x) = e^(-2x).
To find the integrating factor, we use the formula:
μ(x) = e^(∫p(x)dx)
Integrating p(x):
∫[(2x + 1)/x]dx
= ∫(2 + (1/x))dx
= 2x + ln|x| + C1, where C1 is the constant of integration.
Therefore, the integrating factor μ(x) is given by:
μ(x) = e^(2x + ln|x| + C1)
= e^(2x) * e^(ln|x|) * e^(C1)
= e^(2x) * |x| * C2, where C2 is a new constant C1*e^(C1).
Multiplying the given equation by the integrating factor:
e^(2x) * |x| * C2 * dy/dx + e^(2x) * |x| * C2 * ((2x + 1)/x)y = e^(2x) * |x| * C2 * e^(-2x)
Simplifying the equation:
∂/∂x[e^(2x) * |x| * C2 * y] = e^(2x) * |x| * C2 * e^(-2x)
Now, we integrate both sides with respect to x:
∫[∂/∂x(e^(2x) * |x| * C2 * y)]dx = ∫[e^(2x) * |x| * C2 * e^(-2x)]dx
Integrating the left side yields:
e^(2x) * |x| * C2 * y = ∫[e^(2x) * |x| * C2 * e^(-2x)]dx = ∫[C2 * |x|]dx
Integrating the right side:
∫[C2 * |x|]dx = C2 * ∫|x|dx
= C2 * (x * |x|/2 + C3), where C3 is the constant of integration.
Therefore, the previous equation becomes:
e^(2x) * |x| * C2 * y = C2 * (x * |x|/2 + C3)
Dividing by the coefficient of y gives us the final solution:
y = (x * |x|/2 + C3) * e^(-2x) / |x|, where C3/(C2*e^(2x)) is a new constant C4.
Finally, the solution to the differential equation is:
y = (x * |x|/2 + C4) * e^(-2x) / |x|