Question

The differential equation $\dfrac{d^2y}{dx^2} + \dfrac{dy}{dx} + \sin y = 0$ is:

• A. Homogeneous
• B. Non-linear
• C. Linear
• D. of degree two

Hint:

If you see functions like $\sin y$, $e^y$, or $y^2$, the differential equation is almost always non-linear.

Answer 1

The Correct Option is B

Concept: A differential equation is said to be:
• Linear if the dependent variable and its derivatives appear only in first degree and are not multiplied together.
• Non-linear if the equation involves non-linear functions such as $\sin y$, $y^2$, etc.

Step 1: Analyze given equation \[ \frac{d^2y}{dx^2} + \frac{dy}{dx} + \sin y = 0 \] Observe that:
• $\frac{d^2y}{dx^2}$ is linear
• $\frac{dy}{dx}$ is linear
• $\sin y$ is a non-linear function of $y$

Step 2: Identify type Since $\sin y$ is a non-linear function, the equation is non-linear.

Step 3: Reject other options
• Homogeneous: Not applicable here in this context
• Linear: Incorrect due to $\sin y$
• Degree two: Degree refers to highest power of derivatives, here derivatives are of power 1 \[ \therefore \text{Correct answer is (B) Non-linear} \]