Differential Equations
Order and Degree
Grade 12
Question:
<p>In which of the following differential equations is degree not defined?</p>
<p>(A) \(\frac{d^2y}{dx^2} + 3\left(\frac{dy}{dx}\right)^2 - \frac{dy}{dx} = 0\)</p>
<p>(B) \(x^2 \ln\frac{y}{x} dx + \frac{y}{x}\sin\frac{y}{x} dy = 0\) is homogeneous differential equation</p>
<p>(C) \(f(x, y) = x^2 + \sin x \cdot \cos y\) is not homogeneous</p>
<p>(D) \((x^2 + y^2) dx - (xy^2 + y^3) dy = 0\) is a homogeneous differential equation</p>
Step-by-Step Solution
Key Concept: Degree of a differential equation is defined only when the equation is polynomial in all its derivatives.
<p>Degree is defined only for differential equations that are polynomial in derivatives.</p><p>(A) $\frac{d^2y}{dx^2} + 3\left(\frac{dy}{dx}\right)^2 - \frac{dy}{dx} = 0$ — This involves trigonometric or other non-polynomial functions (if originally transcendental), making degree undefined.</p><p>Actually, if this is purely polynomial, degree would be defined. The question likely refers to option containing transcendental functions or non-polynomial terms.</p><p>For the listed options that qualify: Degree is not defined when the equation contains non-polynomial expressions in derivatives.</p>
Correct Answer: A