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Alasa
@Alasa
July 2022
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Проинтегрировать однородное ОДУ
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Удачник66
Verified answer
(x - y)*y dx - x^2 dy = 0
Делим на dx
(x - y)*y - x^2 dy/dx = 0
(x - y)*y - x^2*y' = 0
Однородное уравнение, замена y(x) = x*t(x), тогда y'(x) = t(x) + x*t'(x)
(x - x*t)*x*t - x^2*(t + x*t') = 0
x^2*t - x^2*t^2 - x^2*t - x^3*t' = 0
-x^2*t^2 - x^3*t' = 0
Делим все на -x^2
t^2 + x*t' = 0
x*dt/dx = -t^2
dt/t^2 = -dx/x
1/t = -ln|x| + ln|C| = ln|C/x|
t = x*y(x) = 1/ln|C/x|
y(x) = 1/(x*ln|C/x|
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Answers & Comments
Verified answer
(x - y)*y dx - x^2 dy = 0Делим на dx
(x - y)*y - x^2 dy/dx = 0
(x - y)*y - x^2*y' = 0
Однородное уравнение, замена y(x) = x*t(x), тогда y'(x) = t(x) + x*t'(x)
(x - x*t)*x*t - x^2*(t + x*t') = 0
x^2*t - x^2*t^2 - x^2*t - x^3*t' = 0
-x^2*t^2 - x^3*t' = 0
Делим все на -x^2
t^2 + x*t' = 0
x*dt/dx = -t^2
dt/t^2 = -dx/x
1/t = -ln|x| + ln|C| = ln|C/x|
t = x*y(x) = 1/ln|C/x|
y(x) = 1/(x*ln|C/x|