Question

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Solve the differential equation:
${x}^{2} \frac{ {d}^{2}y }{ {dx}^{2} } + x \frac{dy}{dx} + (x - 1)y = 0$
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Answer 1

Solution

Given , Differential Equation is

(x

2

−yx

2

)dy+(y

2

+xy

2

)dx=0

This can be Simplified as

(yx

2

−x

2

)dy=(y

2

+xy

2

)dx

x

2

(y−1)dy=y

2

(1+x)dx

y

2

dy(y−1)

=

x

2

dx(1+x)

Now On Integrating both side , we get

∫

y

2

dy(y−1)

=∫

x

2

dx(1+x)

∫[

y

1

−

y

2

1

]dy=∫[

x

1

+

x

2

1

]dx

∫[

y

dy

−

y

2

dy

]=∫[

x

dx

+

x

2

dx

]

ln∣y∣−

y

1

=ln∣x∣−

x

1

+lnC

ln∣y∣−ln∣x∣−lnC=

y

1

−

x

1

ln[

c∣x∣

∣y∣

]=

xy

x−y

Answer 2

Answer:

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