Q.(x) Solution of $\frac{dy}{dx}=\frac{y}{x}+\tan\frac{y}{x}$ is $\sin\left(\frac{y}{x}\right)=cx$. (State True or False.)

The key idea is Verification of Solution — substitute the given solution into the differential equation and check if it satisfies the relation.

Step 1: Let $v = \frac{y}{x}$, so $y = vx$ and $\frac{dy}{dx} = v + x\frac{dv}{dx}$. The given solution $\sin v = cx$ implies $x\frac{dv}{dx} = \sin v$ after differentiating:

$\cos v \cdot \frac{dv}{dx} = c \Rightarrow \frac{dv}{dx} = \frac{c}{\cos v}$.

Step 2: From $\sin v = cx$, we have $c = \frac{\sin v}{x}$. Substitute into $\frac{dv}{dx}$:

$\frac{dv}{dx} = \frac{\sin v}{x \cos v} = \frac{\tan v}{x}$. …

The key here is recognising the form of the equation. When you see $\frac{dy}{dx}$ expressed as a function of $\frac{y}{x}$ alone, you're looking at a homogeneous differential equation. The standard trick — substituting $y = vx$ — turns it into a separable equation that can be integrated directly.

Let's walk through it.

1. Rewrite the equation in homogeneous form The given equation is

The right-hand side depends only on the ratio $\frac{y}{x}$, so it is homogeneous of degree zero. This tells us the substitution $y = vx$ (where $v$ is a function of $x$) will work.

2. Substitute $y = vx$ Differentiate:

Also, $\frac{y}{x} = v$. Plugging into the equation:

3. Simplify to a separable form Cancel $v$ on both sides:

This is now separable. Rearrange:

Since $\tan v = \frac{\sin v}{\cos v}$, we can write:

4. Integrate both sides The left side integrates to $\log|\sin v|$ (because the derivative of $\sin v$ is $\cos v$), and the right side integrates to $\log|x| + C$:

5. Solve for the constant Combine the logarithms:

Removing logs (and absorbing the absolute value into the constant $c$): …

Method: Homogeneous Substitution for $\frac{dy}{dx}=\frac{y}{x}+F\!\left(\frac{y}{x}\right)$

When the right side is a function of $\frac{y}{x}$, substitute $v=\frac{y}{x}$; the $\frac{y}{x}$ term cancels, leaving a clean separable equation in $v$.

Steps

Step 1: Substitute $y=vx$.

Step 2: Cancel the $v$ term.

With $\frac{dy}{dx}=v+F(v)$, the equation reduces to $x\frac{dv}{dx}=F(v)$, so $\frac{dv}{F(v)}=\frac{dx}{x}$.

Step 3: Integrate and back-substitute $v=\frac{y}{x}$. …

Common Mistakes

Mistake 1: Not substituting $v=\frac{y}{x}$ and trying to separate directly.

Why it's wrong: $\frac{y}{x}+\tan\frac{y}{x}$ is a function of the ratio, so only the homogeneous substitution untangles it. Correct approach: put $y=vx$, use $\frac{dy}{dx}=v+x\frac{dv}{dx}$, and the $v$ term cancels.

Mistake 2: Integrating $\cot v$ incorrectly. …