UPSC NDA Math Model Paper 7

Given: differential equation is (1−x2)

dy

dx

−xy=1
Divide by (1−x2) on both sides, we get
⇒

dy

dx

−

xy

(1−x2)

=

1

(1−x2)

Now, by comparing the above equation with

dy

dx

+Py=Q So, P=−

x

(1−x2)

Now, Integrating factor =e∫Pdx
=e∫

−x

(1−x2)

dx=e

1

2

‌∫

−2x

(1−x2)

dx
Let 1−x2=tDifferentiating with respect to x, we get
⇒(0−2x)dx=dt
⇒−2xdx=dt
Integrating =e

1

2

‌∫

dt

t

=e

1

2

‌log‌t=elog‌t‌

1

2

=t

1

2

=(1−x2)

1

2

=√1−x2