Q.If $y = (\tan^{-1} x)^2$, show that $(x^2+1)^2 y_2 + 2x(x^2+1) y_1 = 2$. Miscellaneous Examples

Concept: Successive Differentiation — we differentiate the given function twice and substitute into the expression.

Step 1:

$y = (\tan^{-1} x)^2$

First derivative:

$y_1 = 2 \tan^{-1} x \cdot \frac{1}{1+x^2} = \frac{2 \tan^{-1} x}{1+x^2}$

Step 2:

Differentiate $y_1$ using the quotient rule:

$y_2 = \frac{(1+x^2) \cdot \frac{2}{1+x^2} - (2 \tan^{-1} x)(2x)}{(1+x^2)^2} = \frac{2 - 4x \tan^{-1} x}{(1+x^2)^2}$

Step 3:

Multiply $y_2$ by $(x^2+1)^2$ and $y_1$ by $2x(x^2+1)$:

$(x^2+1)^2 y_2 = 2 - 4x \tan^{-1} x$ …

Why This Approach Works

When you see a problem asking you to "show that" a differential relation holds, the natural instinct is to start differentiating directly. But here, $y = (\tan^{-1} x)^2$ — if you differentiate twice in the obvious way, you'll get messy expressions involving $\tan^{-1} x$ and rational functions. The trick is to notice that the derivative of $\tan^{-1} x$ is $\frac{1}{1+x^2}$, which is a clean rational function. So by differentiating once, we get $y_1$ in terms of $\tan^{-1} x$ and $1+x^2$. Then, instead of differentiating $y_1$ directly, we can multiply through by $(1+x^2)$ to simplify the algebra before taking the second derivative.

This is a classic technique: clear the denominator first, then differentiate. It avoids nested fractions and keeps the work tidy.

Step-by-Step Solution

1. Write down the given function and find the first derivative.

We have $y = (\tan^{-1} x)^2$. Differentiate with respect to $x$:

2. Multiply both sides by $(1+x^2)$ to prepare for the next differentiation.

This step is the key insight. Instead of differentiating $y_1$ as a fraction, we write:

Now the right-hand side is just $2\tan^{-1} x$, which is much simpler to differentiate.

3. Differentiate this new equation to find $y_2$.

Differentiate both sides of $(1+x^2) y_1 = 2\tan^{-1} x$ with respect to $x$. Use the product rule on the left:

The right-hand side differentiates to:

So we have: …

Method: Verifying a Relation Involving $(x^2+1)$ Factors (Clear the Denominator First)

Same trick as for logarithmic-argument ODEs: when the first derivative comes out as a fraction, multiply through by the denominator before differentiating a second time, rather than applying the quotient rule directly to a messy fraction.

Steps

Step 1: Find $y_1$

For $y=(\tan^{-1}x)^2$, the chain rule gives $y_1=\dfrac{2\tan^{-1}x}{1+x^2}$.

Step 2: Multiply both sides by $(1+x^2)$ to clear the denominator

$(1+x^2)y_1 = 2\tan^{-1}x$ — now the right-hand side is simple to differentiate. …

Common Mistakes

Mistake 1: Applying the quotient rule directly to $y_1=\dfrac{2\tan^{-1}x}{1+x^2}$ instead of clearing the denominator first.

Why it's wrong: the quotient rule here produces a much messier expression that is far more error-prone to simplify down to the required form. Correct approach: multiply by $(1+x^2)$ first so the next differentiation only needs the product rule on a simple product.

Mistake 2: Forgetting the chain-rule factor $\frac{1}{1+x^2}$ when first differentiating $(\tan^{-1}x)^2$. …