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To solve this differential equation, we'll use an integrating factor. First, we'll rearrange the equation to get y' on its own:
xy' + y = x - 1
y' = (x - 1 - y)/x
Now we'll find the integrating factor, which is e^(integral of x/x dx). The integral of 1/x is ln|x|, so the integrating factor is e^(ln|x|) = |x|.
Multiplying both sides of the equation by the integrating factor gives:
|x|y' + |x|y = |x|(x - 1)
Now we can use the product rule to simplify the left-hand side:
d/dx (|x|y) = |x|y' + y
So the equation becomes:
d/dx (|x|y) = |x|(x - 1)
Integrating both sides with respect to x gives:
|x|y = (1/2)x^2 - x + C
where C is the constant of integration. Solving for y gives:
y = (1/2)x - 1 + C/|x|
where C is the constant of integration.
xy' + y = x - 1
y' = (x - 1 - y)/x
Now we'll find the integrating factor, which is e^(integral of x/x dx). The integral of 1/x is ln|x|, so the integrating factor is e^(ln|x|) = |x|.
Multiplying both sides of the equation by the integrating factor gives:
|x|y' + |x|y = |x|(x - 1)
Now we can use the product rule to simplify the left-hand side:
d/dx (|x|y) = |x|y' + y
So the equation becomes:
d/dx (|x|y) = |x|(x - 1)
Integrating both sides with respect to x gives:
|x|y = (1/2)x^2 - x + C
where C is the constant of integration. Solving for y gives:
y = (1/2)x - 1 + C/|x|
where C is the constant of integration.
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10 апреля 2023 17:33
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