Differential Equations
Differential Equations
star_batch_jee_advanced_2025
Grade 12
Question:
Match the following:
(A) If the function $y = e^{4x} + 2e^{-x}$ is a solution of the differential equation $\frac{d^3y}{dx^3} - 13\frac{dy}{dx} = K$, then the value of $K/3$ is
(B) Number of solutions which satisfy the differential equation $\frac{dy}{dx} + x\left(\frac{dy}{dx}\right)^2 - y = 0$ is
(C) If real value of $m$ for which the substitution, $y = u^m$ will transform the differential equation, $2x^4y\frac{dy}{dx} + y^4 = 4x^6$ into a homogenous equation, then the value of $2m$ is
(D) If the solution of differential equation $x^2\frac{d^2y}{dx^2} + 2x\frac{dy}{dx} = 12y$ is $y = Ax^m + Bx^{-n}$, then $|m - n|$ is
Step-by-Step Solution
Key Concept: Successive differentiation of an exponential sum yields derivatives that satisfy linear differential equations with constant coefficients.
Given $y = e^{4x} + 2e^{-x}$, compute $y_1 = \frac{dy}{dx} = 4e^{4x} - 2e^{-x}$ and $y_2 = \frac{d^2y}{dx^2} = 16e^{4x} + 2e^{-x}$. Then $y_3 = \frac{d^3y}{dx^3} = 64e^{4x} - 2e^{-x}$. Verify that $y_3 - 13y_1 = (64e^{4x} - 2e^{-x}) - 13(4e^{4x} - 2e^{-x}) = 12e^{4x} + 24e^{-x} = 12y$, confirming the differential relation.
Correct Answer: [A-q] [B-r] [C-p] [D-s]