Q.Solve the following differential equation: $\frac{dy}{dx} - \frac{y}{x} + \text{cosec}\left(\frac{y}{x}\right) = 0; y = 0$ when $x = 1$

Homogeneous — the slope is already a function of $y/x$. Put $y=vx$.

Rewrite: $\frac{dy}{dx} = \frac{y}{x} - \csc\left(\frac{y}{x}\right)$.

Substitute $y=vx$, $\frac{dy}{dx}=v+x\frac{dv}{dx}$:

$v + x\frac{dv}{dx} = v - \csc v \implies x\frac{dv}{dx} = -\csc v.$ …

Spotting the type

The equation $\frac{dy}{dx} - \frac{y}{x} + \csc\!\left(\frac{y}{x}\right) = 0$ is written entirely in terms of $y/x$, so it is homogeneous and the substitution $y=vx$ will separate it.

Set up

Substitute $y=vx$

With $\frac{dy}{dx}=v+x\frac{dv}{dx}$,

Separate and integrate

Since $\csc v = \frac{1}{\sin v}$,

$\sin v\,dv = -\frac{dx}{x} \implies -\cos v = -\log|x| + C.$ …

Method: Solving a homogeneous equation by the $y=vx$ substitution

Use this whenever the whole equation can be written so the slope depends only on the ratio $y/x$ — for example a $\csc(y/x)$, $\log(y/x)$ or a ratio of same-degree polynomials.

Steps

Step 1: Confirm it is homogeneous.

Rearrange to $\dfrac{dy}{dx}=F\!\left(\dfrac{y}{x}\right)$. If every remaining piece is built out of $y/x$ alone, the equation is homogeneous and this method is guaranteed to work.

Step 2: Substitute $y=vx$.

Then $\dfrac{dy}{dx}=v+x\dfrac{dv}{dx}$. Replace both $y/x$ (by $v$) and $\dfrac{dy}{dx}$ in the equation.

Step 3: Cancel the stray $v$ and separate. …

Common Mistakes

Mistake 1: Writing $\dfrac{dy}{dx}=v$ instead of $v+x\dfrac{dv}{dx}$ after $y=vx$.

Why it's wrong: $y=vx$ is a product, so $\dfrac{dy}{dx}=v+x\dfrac{dv}{dx}$ by the product rule. Dropping the $x\dfrac{dv}{dx}$ term loses the whole equation. Correct approach: always differentiate $y=vx$ as a product.

Mistake 2: Sign slip when integrating $\int\sin v\,dv$. …