Q.Verify that the function $y = a\cos x + b\sin x$, where $a, b \in \mathbf{R}$ is a solution of the differential equation $\frac{d^2y}{dx^2} + y = 0$.

Concept: Verification of Solution — substitute the given function into the differential equation and check that it satisfies the equation identically.

Step 1: Compute the first derivative:

$y = a\cos x + b\sin x$

$\frac{dy}{dx} = -a\sin x + b\cos x$

Step 2: Compute the second derivative:

$\frac{d^2y}{dx^2} = -a\cos x - b\sin x = -(a\cos x + b\sin x) = -y$

Step 3: Substitute into the differential equation: …

Why this approach works

When we're asked to verify that a given function is a solution of a differential equation, we don't need to solve anything — we just need to check that plugging the function into the equation makes it true. The differential equation here is second-order linear and homogeneous: $\frac{d^2y}{dx^2} + y = 0$. It says, in words, "the second derivative of $y$ plus $y$ itself equals zero for all $x$."

The given function is a combination of sine and cosine. The key insight: both $\sin x$ and $\cos x$ have the property that differentiating them twice brings you back to the negative of the original function. So any linear combination of them will also have that property. That's exactly what the equation demands.

Step-by-step verification

1. Write down the function clearly

We have:

where $a$ and $b$ are any real constants.

2. Find the first derivative

Differentiate term by term:

Remember: derivative of $\cos x$ is $-\sin x$, and derivative of $\sin x$ is $\cos x$.

3. Find the second derivative

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

Here, derivative of $-\sin x$ is $-\cos x$, and derivative of $\cos x$ is $-\sin x$.

4. Observe the pattern

Look at the second derivative:

But $a\cos x + b\sin x$ is exactly $y$! So:

$\frac{d^2y}{dx^2} = -y$ …

Method: Verifying a given function is a solution

Use this whenever a question says "Show that ... is a solution of ..." or "Verify that ...". You are NOT asked to solve the equation — only to confirm that a function already handed to you makes the equation an identity.

Steps

Step 1: List exactly which derivatives the equation contains.

Read the differential equation and note the highest derivative present. If it contains $\dfrac{d^2y}{dx^2}$, you will need $y$, $y'$ and $y''$; if only $\dfrac{dy}{dx}$, you need $y$ and $y'$. Compute no more than you need.

Step 2: Differentiate the given function, treating every constant as a constant.

Constants such as $a$, $b$, $c_1$ are parameters, not functions of $x$ — their derivative is $0$. Differentiate term by term.

Step 3: Substitute $y$ and its derivatives into the left-hand side. …

Common Mistakes

Mistake 1: Differentiating the constants $a$ and $b$.

Why it's wrong: $a$ and $b$ are fixed parameters, not functions of $x$, so they carry through differentiation unchanged. Treating them as variables corrupts every derivative. Correct approach: differentiate only $\cos x$ and $\sin x$, leaving $a$ and $b$ as constant multipliers.

Mistake 2: Sign slips when differentiating trig functions. …