Mister Exam

# Differential equation xydx+(x^2+y^2)dy=0

y() =
y'() =
y''() =
y'''() =
y''''() =

from to

### The solution

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          2 d           2    d
x*y(x) + x *--(y(x)) + y (x)*--(y(x)) = 0
dx               dx          
$$x^{2} \frac{d}{d x} y{\left(x \right)} + x y{\left(x \right)} + y^{2}{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = 0$$
x^2*y' + x*y + y^2*y' = 0
            _____________________
/           _________
/     2     /       4
y(x) = -\/   - x  - \/  C1 + x   
$$y{\left(x \right)} = - \sqrt{- x^{2} - \sqrt{C_{1} + x^{4}}}$$
           _____________________
/           _________
/     2     /       4
y(x) = \/   - x  - \/  C1 + x   
$$y{\left(x \right)} = \sqrt{- x^{2} - \sqrt{C_{1} + x^{4}}}$$
            ___________________
/    _________
/    /       4     2
y(x) = -\/   \/  C1 + x   - x  
$$y{\left(x \right)} = - \sqrt{- x^{2} + \sqrt{C_{1} + x^{4}}}$$
           ___________________
/    _________
/    /       4     2
y(x) = \/   \/  C1 + x   - x  
$$y{\left(x \right)} = \sqrt{- x^{2} + \sqrt{C_{1} + x^{4}}}$$
Graph of the Cauchy problem
The classification
1st exact
1st homogeneous coeff best
1st homogeneous coeff subs indep div dep
1st homogeneous coeff subs dep div indep
1st power series
lie group
1st exact Integral
1st homogeneous coeff subs indep div dep Integral
1st homogeneous coeff subs dep div indep Integral
(x, y):
(-10.0, 0.75)
(-7.777777777777778, 0.9619689990676026)
(-5.555555555555555, 1.3328538606771176)
(-3.333333333333333, 2.063993497553178)
(-1.1111111111111107, 3.075458642648265)
(1.1111111111111107, 3.075458760354001)
(3.333333333333334, 2.063993667128847)
(5.555555555555557, 1.3328537437352466)
(7.777777777777779, 0.9619688864903184)
(10.0, 0.749999949196909)
(10.0, 0.749999949196909)
The graph
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