Q.If $y = A\sin x + B\cos x$, then prove that $\frac{d^2y}{dx^2} + y = 0$.

Concept: Second-order linear differential equation with constant coefficients — verifying a given solution.

We are given $y = A\sin x + B\cos x$, where $A$ and $B$ are constants.

Step 1: Differentiate once with respect to $x$:

Step 2: Differentiate again:

Step 3: Notice that the right-hand side is exactly $- (A\sin x + B\cos x) = -y$.

Therefore:

Why this works: the core idea

The equation $\frac{d^2y}{dx^2} + y = 0$ is the simple harmonic oscillator differential equation. Its general solution is exactly $y = A\sin x + B\cos x$. So the problem is essentially asking you to verify that the given function satisfies the equation it was designed to solve.

The key insight: differentiating $\sin x$ twice gives $-\sin x$, and differentiating $\cos x$ twice gives $-\cos x$. So the second derivative just flips the sign of each term, producing $-y$.

Step-by-step verification

1. Write down the given function

We have $y = A\sin x + B\cos x$, where $A$ and $B$ are constants.

2. Find the first derivative

Differentiate term by term:

• Derivative of $\sin x$ is $\cos x$
• Derivative of $\cos x$ is $-\sin x$

So:

3. Find the second derivative

Differentiate $\frac{dy}{dx}$:

• Derivative of $\cos x$ is $-\sin x$
• Derivative of $-\sin x$ is $-\cos x$

So:

4. Observe the pattern

Notice that $-A\sin x - B\cos x$ is exactly $-(A\sin x + B\cos x)$, which is $-y$.

Therefore:

5. Rearrange to get the required form

Adding $y$ to both sides:

Method: Verifying That a Function Satisfies a Given Differential Equation

This method applies whenever you are asked to prove or show that a given $y$ satisfies a relation involving $y$ and its derivatives (rather than solve the equation from scratch).

Steps

Step 1: Differentiate the given $y$ twice

Find $\dfrac{dy}{dx}$ first, then differentiate that result again to get $\dfrac{d^2y}{dx^2}$, using the standard derivative rules term by term.

Step 2: Substitute every derivative into the given expression

Write out the left-hand side of the relation to be proved (e.g. $\dfrac{d^2y}{dx^2}+y$) using the exact expressions found in Step 1 — do not skip writing $y$ itself back in.

Step 3: Simplify algebraically until the expression collapses to the required value

Combine like terms (grouping $\sin x$ terms together and $\cos x$ terms together, for functions built from $A\sin x+B\cos x$) and show the sum reduces exactly to $0$ (or whatever the target of the proof is) — this confirms the identity rather than assuming it.

Common Mistakes

Mistake 1: Sign slip differentiating $\cos x$ or $-\sin x$

Why it's wrong: $\dfrac{d}{dx}(\cos x)=-\sin x$ and $\dfrac{d}{dx}(-\sin x)=-\cos x$ — two consecutive sign changes are easy to lose track of, especially with $A$ and $B$ also present. Correct approach: differentiate one term at a time and write the sign explicitly at each step rather than tracking it mentally across both derivatives.

Mistake 2: Losing the coefficients $A$ and $B$ partway through

Why it's wrong: dropping or swapping $A$ and $B$ midway (e.g. attaching $A$ to the $\cos x$ term instead of $\sin x$) breaks the cancellation that makes the proof work. Correct approach: keep $A$ and $B$ attached to their original terms ($A$ with $\sin x$, $B$ with $\cos x$) through every differentiation step.