Question Statement

Verify that the function $y=a\cos x+b\sin x$, where $a,b\in R$, is a solution of the differential equation:

$\frac{d^{2}y}{d{x}^2}+y=0$

---

Background and Explanation

To verify that a given function is a solution to a differential equation, we must calculate the derivatives of the function required by the equation and substitute them back into it. If the substitution results in an identity (where the left-hand side equals the right-hand side), the function is a valid solution.

---

Solution

Given the function: $y=a\cos x+b\sin x$

And the differential equation to verify: $\frac{d^{2}y}{d{x}^2}+y=0$

Step 1: Find the first derivative

Differentiate the function $y$ with respect to $x$: $\frac{dy}{dx}=\frac{d}{dx}(a\cos x+b\sin x)$ $\frac{dy}{dx}=a(-\sin x)+b(\cos x)$ $\frac{dy}{dx}=-a\sin x+b\cos x$

Step 2: Find the second derivative

Differentiate $\frac{dy}{dx}$ again with respect to $x$ to find $\frac{d^{2}y}{d{x}^2}$: $\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}(-a\sin x+b\cos x)$ $\frac{d^{2}y}{d{x}^2}=-a(\cos x)+b(-\sin x)$ $\frac{d^{2}y}{d{x}^2}=-a\cos x-b\sin x$

Step 3: Substitute into the differential equation

Now, substitute the values of $y$ and $\frac{d^{2}y}{d{x}^2}$ into the left-hand side (LHS) of equation (1): $\text{LHS}=\frac{d^{2}y}{d{x}^2}+y$ $\text{LHS}=(-a\cos x-b\sin x)+(a\cos x+b\sin x)$

Group the like terms: $\text{LHS}=(-a\cos x+a\cos x)+(-b\sin x+b\sin x)$ $\text{LHS}=0+0$ $0=0$

Since the LHS equals the RHS, the function $y=a\cos x+b\sin x$ is indeed a solution of the differential equation $\frac{d^{2}y}{d{x}^2}+y=0$.

---

Key Formulas or Methods Used

• Derivative of Sine: $\frac{d}{dx}(\sin x)=\cos x$
• Derivative of Cosine: $\frac{d}{dx}(\cos x)=-\sin x$
• Verification Method: Direct substitution of the function's derivatives into the differential equation.

---

Summary of Steps

1. Write down the given function $y$ and the differential equation.
2. Compute the first derivative $\frac{dy}{dx}$ using standard trigonometric derivative rules.
3. Compute the second derivative $\frac{d^{2}y}{d{x}^2}$ by differentiating the first derivative.
4. Substitute the expressions for $y$ and $\frac{d^{2}y}{d{x}^2}$ into the differential equation.
5. Simplify the expression to show that $0=0$, confirming the solution.