I have a problem with differential equations. "Solve the initial value problem: dy/dx +2y = 4x, y(0) = 2" If you can help, even if it's only pointing me in the right direction for what process I need to use, that'd be great.
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This is called the method of integrating factors.
The idea is that we'd like to rewrite the Left Hand Side of
y' +f(x) y = g(x)
as the derivative of a product of y with some other function h(x).
Notice that the derivative of y(x)h(x) is y’h + h’y.
This kind of looks like y’ + f(x)*y.
But to make that look even more like y’h + h’y,
we can multiply y’ + f(x)*y by the function h (to be determined)
to get h[y’ + f(x)*y] = y’h + (h*f)*y.
So what should our function h be ?
Our h should satisfy (hf) = h’ to match the expressions y’h + (hf)*y and y’h + h’y.
Solving hf = h’ is easy since its separable.
In this case, if we have y’ + 2y = x^2
So we’d like to multiply both sides of the equation by some h(x).
And that h(x) should satisfy h*(2) = h’.
Solving this gives h(x) = e^(2x)
Then multiply both sides of y’ + 2y = 4x by e^(2x),
recognize that the Left Hand Side is the derivative of y*e^(2x),
and continue from there. Once you get the general solution, plug in the initial values to get the specified solution.
Thanks a lot verifry.
I had some notes that showed it, but the process was completely lost on me. A massive help.