JEE Main 2026 — Mathematics Calculus

The given differential equation is:

$$x\sqrt{1-x^2}\,dy + \left(y\sqrt{1-x^2} - x\cos^{-1}x\right)dx = 0$$x1−x2​dy+(y1−x2​−xcos−1x)dx=0

Rearranging the terms, we get:

$$x\sqrt{1-x^2}\,dy + y\sqrt{1-x^2}\,dx = x\cos^{-1}x\,dx$$x1−x2​dy+y1−x2​dx=xcos−1xdx

Dividing the entire equation by $\sqrt{1-x^2}$1−x2​:

$$x\,dy + y\,dx = \dfrac{x\cos^{-1}x}{\sqrt{1-x^2}}\,dx$$xdy+ydx=1−x2​xcos−1x​dx

The left side is the exact differential of $xy$xy:

$$d(xy) = \dfrac{x\cos^{-1}x}{\sqrt{1-x^2}}\,dx$$d(xy)=1−x2​xcos−1x​dx

Integrating both sides:

$$xy = \int \dfrac{x\cos^{-1}x}{\sqrt{1-x^2}}\,dx + C$$xy=∫1−x2​xcos−1x​dx+C

To evaluate the integral, use integration by parts. Let $u = \cos^{-1}x$u=cos−1x and $dv = \dfrac{x}{\sqrt{1-x^2}}\,dx$dv=1−x2​x​dx.

$du = -\dfrac{1}{\sqrt{1-x^2}}\,dx$du=−1−x2​1​dx and $v = -\sqrt{1-x^2}$v=−1−x2​

$$\int u\,dv = uv - \int v\,du$$∫udv=uv−∫vdu

$$\int \dfrac{x\cos^{-1}x}{\sqrt{1-x^2}}\,dx = -\sqrt{1-x^2}\cos^{-1}x - \int \left(-\sqrt{1-x^2}\right)\left(-\dfrac{1}{\sqrt{1-x^2}}\right)\,dx$$∫1−x2​xcos−1x​dx=−1−x2​cos−1x−∫(−1−x2​)(−1−x2​1​)dx

$$= -\sqrt{1-x^2}\cos^{-1}x - \int 1\,dx$$=−1−x2​cos−1x−∫1dx

$$= -\sqrt{1-x^2}\cos^{-1}x - x$$=−1−x2​cos−1x−x

Substituting this back into the equation:

$$xy = -\sqrt{1-x^2}\cos^{-1}x - x + C$$xy=−1−x2​cos−1x−x+C

It is given that $\displaystyle \lim_{x\to 1^-} y(x) = 1$x→1−lim​y(x)=1. Substituting $x = 1$x=1 and $y = 1$y=1:

$$1(1) = -\sqrt{1-1^2}\cos^{-1}(1) - 1 + C$$1(1)=−1−12​cos−1(1)−1+C

$$1 = 0 - 1 + C \Rightarrow C = 2$$1=0−1+C⇒C=2

The particular solution is:

$$xy = 2 - x - \sqrt{1-x^2}\cos^{-1}x$$xy=2−x−1−x2​cos−1x

To find $y\left(\dfrac{1}{2}\right)$y(21​), substitute $x = \dfrac{1}{2}$x=21​:

$$\dfrac{1}{2}y\left(\dfrac{1}{2}\right) = 2 - \dfrac{1}{2} - \sqrt{1 - \left(\dfrac{1}{2}\right)^2}\cos^{-1}\left(\dfrac{1}{2}\right)$$21​y(21​)=2−21​−1−(21​)2​cos−1(21​)

$$\dfrac{1}{2}y\left(\dfrac{1}{2}\right) = \dfrac{3}{2} - \dfrac{\sqrt{3}}{2}\left(\dfrac{\pi}{3}\right)$$21​y(21​)=23​−23​​(3π​)

$$\dfrac{1}{2}y\left(\dfrac{1}{2}\right) = \dfrac{3}{2} - \dfrac{\pi}{2\sqrt{3}}$$21​y(21​)=23​−23​π​

Multiplying by $2$2:

$$y\left(\dfrac{1}{2}\right) = 3 - \dfrac{\pi}{\sqrt{3}}$$y(21​)=3−3​π​