Differential Equations
Differential Equations
star_batch_jee_advanced_2025
Grade 12
Question:
Match the following:
(A) Order of differential equation whose general solution is given by $y = (c_1 + c_2)\cos(x + c_3) - c_4e^{x+c_5} + c_4 \sin x$, where $c_1, c_2, c_3, c_4, c_5$ are arbitrary constants, is
(B) Order of differential equation formed by eliminating the constants from $y = a\sin^2 x + b\cos^2 x + c\sin 2x + d \cos 2x + e\sin x$, where $a, b, c, d$ are arbitrary constants, is
(C) The degree of equation $\frac{d^2y}{dx^2} - \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/2} = 0$
(D) Order of differential equation whose solution is $y = cx + c^2 - 3c^{3/2} + 2$, where $c$ is a parameter is
Step-by-Step Solution
Key Concept: Identifying the order and degree of differential equations through differentiation and recognizing standard solution forms.
Part (A) recognizes that $y = d\cos(x + b) + ce^x + d\sin x$ is a solution by direct differentiation and comparison with a differential equation. Part (B) identifies $y = a(\frac{1-\cos 2x}{2}) + b(\frac{1+\cos 2x}{2}) + e\sin x + d\cos 2x + e\sin x$ as a particular form. Part (C) shows that $[1 + (\frac{dy}{dx})^2] = [\frac{d^2y}{dx^2}]^2$ is a second-order differential equation characteristic of certain curves. Part (D) states that upon differentiation, $\frac{dy}{dx} = c$, indicating that the order of the differential equation is 1.
Correct Answer: [A-r] [B-s] [C-q] [D-p]