Question

Which one of the following is a non-linear, second-order, and first-degree differential equation?

• A. \( \dfrac{d^2y}{dx^2} + \cos x \dfrac{dy}{dx} + e^x y = 0 \)
• B. \( \dfrac{d^2y}{dx^2} + \cos y = 0 \)
• C. \( \dfrac{d^2y}{dx^2} + \sin x = 0 \)
• D. \( \left(\dfrac{dy}{dx}\right)^2 + 3y = 0 \)

Hint:

Order comes from the highest derivative present, degree from its power, and linearity fails only when y itself sits inside a non-linear function.

Answer 1

The Correct Option is B

This question checks three ideas about a differential equation at once: its order (how many times y is differentiated), its degree (the power on the highest derivative once the equation is a polynomial in derivatives), and its linearity (whether y and its derivatives appear only in first power, with coefficients depending on x alone).

1. Option (A): $\frac{d^2y}{dx^2} + \cos x \frac{dy}{dx} + e^x y = 0$. The highest derivative is the second derivative, so order 2, and it appears to the power 1, so degree 1. The multipliers $\cos x$ and $e^x$ depend only on $x$, and $y$ is not wrapped inside any non-linear function, so this equation is linear. It fails the non-linear requirement.
2. Option (B): $\frac{d^2y}{dx^2} + \cos y = 0$. Order is 2 and degree is 1, same as before. The catch is $\cos y$: cosine of the dependent variable itself is a non-linear function of $y$, so the whole equation becomes non-linear. This matches all three conditions together.
3. Option (C): $\frac{d^2y}{dx^2} + \sin x = 0$. The sine here acts on $x$, the independent variable, not on $y$. A function of $x$ alone is just an ordinary term, so the equation stays linear. Order 2, degree 1, but linear, so it fails.
4. Option (D): $\left(\frac{dy}{dx}\right)^2 + 3y = 0$. The highest derivative present is only the first derivative, so order is 1, not 2. Since that first derivative is squared, the degree is 2. This equation is non-linear, but first order and second degree, not second order and first degree.

Checking each option against all three required properties together (non-linear and second order and first degree) leaves only option (B) standing.

Let's summarize:

• Order comes from counting how many times y is differentiated: A, B, C all have order 2, while D has order 1.
• Degree comes from the power on that highest derivative: A, B, C have degree 1, while D has degree 2.
• Linearity fails only when y itself sits inside a non-linear function like $\cos y$; that happens only in option B.

So the differential equation in option (B), $\frac{d^2y}{dx^2} + \cos y = 0$, is non-linear, second order, and first degree.