y=(x−y dx/dy)sin(x/y), x(1)=π/2 — Find cos(x(2))

JEE Main 2025 — 23 January, Evening Shift. Previous Year Question.

Problem

Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y\dfrac{dx}{dy}\right)\sin\left(\dfrac{x}{y}\right)$, where $y$ is positive and $x(1)=\dfrac{\pi}{2}$. Then $\cos(x(2))$ is equal to:

Key insight. The equation mixes $x$ and $y$ only through the ratio $\dfrac{x}{y}$ — a strong signal to substitute $v=\dfrac{x}{y}$. Once $x=vy$ is substituted throughout (treating $x$ as a function of $y$), the messy differential equation collapses into a simple separable one in $v$ and $y$.

Watch this explained step by step →  ·  More on the Shiwam’s Classes channel

Approach

Substitute $v=\dfrac{x}{y}$, so $x=vy$ and $\dfrac{dx}{dy}=v+y\dfrac{dv}{dy}$. Rewrite the differential equation entirely in terms of $v$ and $y$, separate the variables, and integrate. Use the initial condition to find the constant of integration, then evaluate the resulting relation at $y=2$ and use the double-angle formula for cosine to find $\cos(x(2))$.

Solution

Step 1 — Substitute v = x/y

With $x=vy$: $\dfrac{dx}{dy}=v+y\dfrac{dv}{dy}$, so:

$$x-y\frac{dx}{dy} = vy-y\left(v+y\frac{dv}{dy}\right) = vy-vy-y^2\frac{dv}{dy} = -y^2\frac{dv}{dy}$$

Step 2 — Rewrite the differential equation

$$y = \left(-y^2\frac{dv}{dy}\right)\sin(v)$$

Dividing both sides by $y$ (valid since $y>0$):

$$1 = -y\sin(v)\frac{dv}{dy}$$

Step 3 — Separate variables

$$\sin(v)\,dv = -\frac{dy}{y}$$

Step 4 — Integrate both sides

$$\int\sin(v)\,dv = -\int\frac{dy}{y}$$

$$-\cos(v) = -\ln y + C \implies \cos(v) = \ln y + C’$$

(renaming the constant for convenience)

Step 5 — Apply the initial condition

At $y=1$, $x=\dfrac{\pi}{2}$, so $v=\dfrac{x}{y}=\dfrac{\pi}{2}$:

$$\cos\left(\frac{\pi}{2}\right) = \ln(1) + C’ \implies 0 = 0+C’ \implies C’=0$$

Step 6 — Write the simplified solution

$$\cos(v) = \ln y \implies \cos\left(\frac{x}{y}\right) = \ln y$$

Step 7 — Evaluate at y = 2

$$\cos\left(\frac{x(2)}{2}\right) = \ln 2$$

Step 8 — Use the double-angle formula to find cos(x(2))

Let $X=x(2)$. Since $\cos\left(\dfrac{X}{2}\right)=\ln2$, and $\cos(X)=2\cos^2\left(\dfrac{X}{2}\right)-1$:

$$\cos(X) = 2(\ln2)^2-1$$

Answer

Common mistakes

• Trying to solve the equation directly in $x$ and $y$ without the substitution. The mixed term $\sin\left(\frac{x}{y}\right)$ makes the equation impossible to separate directly — the $v=\frac{x}{y}$ substitution is essential.
• Forgetting to apply the double-angle formula at the end. The relation found is $\cos\left(\frac{x(2)}{2}\right)=\ln2$, not $\cos(x(2))=\ln2$ directly — the final step of converting a half-angle cosine to a full-angle cosine is easy to skip.

Practise next

• Let $x=x(y)$ solve $y=\left(x-y\dfrac{dx}{dy}\right)\cos\left(\dfrac{x}{y}\right)$ with $x(1)=0$. Find $x(e)$, using the same substitution method.