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Please solve the following differential equations:?
(1)x^2(x^2-1)dy/dx+x(x^2+1)y=x^2-1
(2)xcos(y/x)(ydx+xdy)=ysin(y/x)(xdy-ydx)
2 Answers
- Tabula RasaLv 61 decade agoFavorite Answer
(2) xcos(y/x) (ydx + xdy) = ysin(y/x) (xdy - ydx) | : xcos(y/x) ≠ 0
y dx + x dy = [y/x]·[(sin(y/x))/(cos(y/x)) · (x dy - y dx)
d(xy) = [y/x]·[(sin(y/x))/(cos(y/x)) · [ - (y dx - x dy)/x²] · x²
d(xy) = – x² · [y/x]·[(sin(y/x))/(cos(y/x)) · d(y/x)
d(xy) = – (xy) [(sin(y/x))/(cos(y/x)) · d(y/x) | : xy ≠ 0
[d(xy)]/(xy) = – [(sin(y/x))/(cos(y/x)) · d(y/x)
∫ [d(xy)]/(xy) = – ∫ [(sin(y/x))/(cos(y/x)) · d(y/x)
ln| xy| = ∫ [1/(cos(y/x))] d(cos(y/x))
ln|xy| = ln| cos(y/x)| + ln|C|
ln|xy| = ln|Ccos(y/x)|
xy = C·cos(y/x)
xy - C·cos(y/x) = 0 <----------------------- ANSWER
(1) x²(x² - 1)dy/dx + x(x²+1)y = x² - 1 | : x²(x² - 1) ≠ 0
y' + [(x² + 1)/(x(x² - 1))] ·y = 1/x²
It is the linear first order differential equation
http://tutorial.math.lamar.edu/Classes/DE/Linear.a...
Good luck!
