Q.Solve the following differential equation: $\frac{dy}{dx} = \sqrt{4 - y^2}$ ($-2 < y < 2$)

Concept: Separation of Variables — isolate $y$ and $x$ on opposite sides, then integrate.

Step 1: Separate variables.

Step 2: Integrate both sides.

The left integral is a standard form: $\int \frac{dy}{\sqrt{a^2 - y^2}} = \sin^{-1}\left(\frac{y}{a}\right) + C$. Here $a = 2$, so

Step 3: Solve for $y$.

The condition $-2 < y < 2$ is automatically satisfied for real $x$.

The key idea here is Separation of Variables. When a differential equation can be written in the form $\frac{dy}{dx} = f(x)g(y)$, we can rearrange it so that all $y$ terms (including $dy$) are on one side and all $x$ terms (including $dx$) are on the other. Then we integrate both sides. This works because we are essentially treating $dy$ and $dx$ as differentials that can be moved algebraically — a standard technique in first-order ODEs.

Our equation is $\frac{dy}{dx} = \sqrt{4 - y^2}$. Notice the right-hand side depends only on $y$, not on $x$. That makes it a special case of separable: $f(x) = 1$ and $g(y) = \sqrt{4 - y^2}$.

Let’s solve it step by step.

1. Separate the variables. Multiply both sides by $dx$ and divide by $\sqrt{4 - y^2}$:

The left side is now purely in $y$, the right side purely in $x$.

2. Integrate both sides.

The right-hand integral is straightforward: $\int dx = x + C$, where $C$ is an arbitrary constant.

3. Evaluate the left-hand integral. This is a standard form. Recall:

Here $a = 2$ (since $4 = 2^2$). So:

We can absorb $C_1$ into the constant from the other side.

4. Combine the results. Equating the two integrals (and merging constants):

where $C$ is an arbitrary constant.

5. Solve for $y$ explicitly. Take the sine of both sides:

Multiply by 2:

This is the general solution. The constant $C$ is determined by an initial condition if one is given. The domain restriction $-2 < y < 2$ is automatically satisfied for all real $x$ because $|\sin(x+C)| \leq 1$, so $|y| \leq 2$, and equality only at isolated points (where the derivative would be zero, but the original ODE still holds in the limit).

Method: Separation with a standard inverse-trig integral

Use this when $\dfrac{dy}{dx}=g(y)$ (right side depends only on $y$) and $g(y)$ matches a standard integral form such as $\sqrt{a^2-y^2}$.

Steps

Step 1: Separate

Divide by $g(y)$ and multiply by $dx$:

Step 2: Recognise the standard form

Identify $a$ by matching the constant under the root (here $4=2^2$ gives $a=2$).

Step 3: Integrate and solve explicitly

Integrate both sides, then invert to get $y$:

Common Mistakes

Mistake 1: Misremembering the standard integral

Why it's wrong: $\int\dfrac{dy}{\sqrt{a^2-y^2}}=\sin^{-1}\dfrac{y}{a}$, not $\cos^{-1}$ or a logarithm; using the wrong form gives a wrong solution shape. Correct approach: match $\sqrt{4-y^2}$ to $a=2$ and use the inverse-sine form.

Mistake 2: Ignoring the domain $-2<y<2$

Why it's wrong: the separated form $\dfrac{dy}{\sqrt{4-y^2}}$ is undefined at $y=\pm2$, so the manipulation is only valid strictly inside the interval. Correct approach: state the restriction and keep $|y|<2$.

Mistake 3: Leaving the answer as $\sin^{-1}\frac{y}{2}=x+C$ when an explicit form is wanted

Why it's wrong: an incomplete inversion misses the expected $y=2\sin(x+C)$. Correct approach: take sine of both sides to solve for $y$.