Question

Find the particular solution of the differential equation \(\dfrac{dy}{dx}+y\cot x=4x\csc x\), \((x\neq0)\), given that \(y=0\) when \(x=\dfrac{\pi}{2}\).

Hint:

This is linear in y. Use integrating factor \(\sin x\), simplify \(4x\csc x\cdot\sin x\), then apply the given condition.

Answer 1

Step 1: Understanding the Concept:
The equation \(\dfrac{dy}{dx}+y\cot x=4x\csc x\) is a linear differential equation in y, of the form \(\dfrac{dy}{dx}+Py=Q\) with \(P=\cot x\) and \(Q=4x\csc x\).
Such equations are solved using an integrating factor.

Step 2: Find the integrating factor:
\[ I.F.=e^{\int\cot x\,dx}=e^{\ln|\sin x|}=\sin x \]

Step 3: Write and integrate the solution:
Multiply both sides by \(\sin x\); the left side becomes the derivative of \(y\sin x\).
\[ \frac{d}{dx}(y\sin x)=4x\csc x\cdot\sin x=4x \]
Integrate both sides with respect to x.
\[ y\sin x=\int4x\,dx=2x^2+C \]

Step 4: Apply the initial condition:
Put \(x=\dfrac{\pi}{2}\), \(y=0\).
\[ 0\cdot\sin\frac{\pi}{2}=2\left(\frac{\pi}{2}\right)^2+C \implies 0=\frac{\pi^2}{2}+C \implies C=-\frac{\pi^2}{2} \]

Final Answer:
Substitute C back to get the particular solution. \[ \boxed{y=\frac{4x^2-\pi^2}{2\sin x}} \]