Q.Solve the following differential equation: $y - \cos y = x$ ; $(y \sin y + \cos y + x) y' = y$

Concept: Implicit Differentiation — we differentiate the given relation with respect to $x$ and then compare with the given differential equation.

Step 1: Differentiate $y - \cos y = x$ implicitly with respect to $x$:

Step 2: From the above, $y' = \frac{1}{1 + \sin y}$. Substitute this into the given differential equation:

Step 3: Multiply both sides by $1 + \sin y$:

Step 4: Cancel $y \sin y$ from both sides, leaving: …

Why implicit differentiation?

We are given two pieces: an implicit relation between $x$ and $y$,

and a differential equation,

The natural question: does the first relation satisfy the second? Since $y$ is not isolated (it appears both inside and outside the cosine), we cannot write $y$ as an explicit function of $x$ in elementary terms. That’s exactly where implicit differentiation shines — we differentiate both sides of the relation with respect to $x$, treating $y$ as a function of $x$, and then compare the result with the given differential equation.

Step-by-step solution

1. Differentiate the implicit relation

We start with

Differentiate both sides with respect to $x$:

The derivative of $y$ is $y'$. For $\cos y$, we use the chain rule: derivative of $\cos y$ is $-\sin y$, times $y'$. So:

That simplifies to:

2. Factor and solve for $y'$

Factor $y'$ out of the left side:

Hence:

3. Substitute into the differential equation

The given differential equation is:

Replace $y'$ with $\frac{1}{1 + \sin y}$:

4. Use the original relation to simplify $x$

We know from the relation that $x = y - \cos y$. Substitute this into the bracket: …

Method: Verify an implicit solution, using the relation to remove $x$

Use this when a relation like $y-\cos y=x$ must be checked against an equation that also contains $x$.

Steps

Step 1: Differentiate the relation implicitly and solve for $y'$.

Differentiating $y-\cos y=x$ gives $y'(1+\sin y)=1$, so $y'=\frac{1}{1+\sin y}$.

Step 2: Substitute $y'$ into the target equation.

Replace $y'$ in $(y\sin y+\cos y+x)\,y'=y$. …

Common Mistakes

Mistake 1: Leaving $x$ in the expression.

Why it's wrong: after substituting $y'=\frac{1}{1+\sin y}$, the bracket $y\sin y+\cos y+x$ cannot simplify to $y(1+\sin y)$ unless you replace $x$ using the given relation $x=y-\cos y$. Correct approach: use the original relation to eliminate $x$.

Mistake 2: Forgetting the chain-rule sign on $\cos y$. …