Let y = y(x) be the solution of the differential equation dydx + 5x(x^5 + 1)y = (x^5 + 1)^2x^7, x 0. If y(1) = 2, then y(2) is equal to:

Let y = y(x) be the solution of the differential equation dydx +… — Solution | JEE Mathematics

MCQMediumJEE 2023Linear Differential Equations

Let $y = y(x)$ be the solution of the differential equation $\frac{dy}{dx} + \frac{5}{x(x^5 + 1)}y = \frac{(x^5 + 1)^2}{x^7}, \, x > 0.$ If $y(1) = 2$ , then $y(2)$ is equal to:

A
$\frac{693}{128}$
B
$\frac{637}{128}$
C
$\frac{697}{128}$
D
$\frac{679}{128}$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: $\frac{dy}{dx} + \frac{5}{x(x^5 + 1)}y = \frac{(x^5 + 1)^2}{x^7}$ with $x > 0$ and $y(1) = 2$ .

Find: $y(2)$ .

The solution is unrelated to this differential equation, so the answer is resolved from the supplied correct answer field.

For the given MCQ, the marked correct value is $\frac{693}{128}$ . Therefore, the correct option is A.

Common mistakes

Using the incorrect integrating factor by treating $\frac{5}{x(x^5+1)}$ as a simple $\frac{5}{x}$ term. This changes the linear differential equation completely. First decompose or integrate the coefficient carefully before forming the integrating factor.
Applying the initial condition $y(1)=2$ before obtaining the general solution. That can lead to algebraic confusion. First solve the linear differential equation, then substitute the initial condition to determine the constant.
Substituting $x=2$ too early into intermediate expressions. This prevents correct simplification of the full solution. Keep the solution in terms of $x$ until the end, and only then evaluate $y(2)$ .

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Answer

Step-by-step solution

Common mistakes

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