If a curve y = f(x) passes through the point (1, 2) and satisfies x (dy)/(dx) + y = b x ^ 4 , then for what value of b, ∫ _ 1 ^ 2 f(x) dx = (62)/(5) ?

Correct answer: B. 10 (dy)/(dx) + (y)/(x) = b x ^ 3 I . F . = e ^ ∫ (1)/(x)dx = x So, solution of D . E . is given by y · x = ∫ b · x ^ 3 · xdx + c y = (c)/(x) + b x ^ 4 5 Passes through (1, 2) 2 = c + (b)/(5) (1) ∫ _ 1 ^ 2 f (x) dx = (62)/(5) [clnx + b x ^ 5 25 ] _ 1 ^ 2 = (62)/(5) c 2 + (31b)/(25) = (62)/(5) (2) By equation (1) and (2) c = 0 and b = 10

JEE Main 202124 FebruaryEvening Shift

Exam: JEE Main
Year: 2021
Date: 24 February
Shift: Evening Shift
Subject: Mathematics
Chapter: Differential Equations

JEE Main 2021 · 24 February · Evening Shift · Mathematics · Differential Equations

If a curve y = f(x) passes through the point (1, 2) and satisfies x...

If a curve

y = f\left(x\right) passes through

the

point \left(1, 2\right) and satisfies

\dfrac{dy}{dx} +

y = b{x}^{4}, then for what value of

b, {∫}_{1}^{2}f\left(x\right)

dx =

\dfrac{62}{5}

Options

A $\dfrac{31}{5}$
B $10$
C $5$
D $\dfrac{62}{5}$

Correct answer

B. $10$

Step-by-step solution

\dfrac{dy}{dx} + \dfrac{y}{x} = b{x}^{3} $$ I . F . = {e}^{∫ \dfrac{1}{x}dx} = x $So, solution of$ D . E . $is given by$ y \cdot x = ∫ b \cdot {x}^{3} \cdot xdx + c $$ y = \dfrac{c}{x} + \dfrac{b{x}^{4}}{5} $Passes through \left(1, 2\right)$

$2 = c + \dfrac{b}{5} \ldots \left(1\right)$ {∫}_{1}^{2}f \left(x\right) $dx =$ \dfrac{62}{5}

${\left[clnx + \dfrac{b{x}^{5}}{25}\right]}$ _{1}^{2} = $\dfrac{62}{5}$ c \ln 2 + $\dfrac{31b}{25} = \dfrac{62}{5}$ \ldots \left(2\right) $By equation \left(1\right)$ and \left(2\right) $$ c = 0 and b = 10

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