If a curve y = f(x) passes through the point (1, 2) and satisfies xdy/dx + y = bx⁴, then for what value of b,

${\int }_1^{2}f(x)dx=\frac{62}{5}$

Concept:

The standard form of a First-order Linear Differential Equation is $\frac{dy}{dx}+Py=Q$, where P and Q are the functions of x.

The solution for the above differential equation is $y\left(IF\right)=\int (IF\times Q)dx+C$, where IF is the integrating factor $IF=e^{\int Pdx}$

Calculation:

Given, xdy/dx + y = bx4

⇒ dy/dx + y/x = bx3

∴ I.F. = $e^{\int \frac{dx}{x}}$  e∫dxx = x

∴ yx = ∫bx4 dx = bx5/5 + c

Now, the above curve passes through (1, 2).

⇒ 2 = b/5 + c

Also,

${\int }_1^2(\frac{b{x}^4}{5}+\frac{c}{x})dx=\frac{62}{5}$

⇒ (b/25) × 32 + c ln 2 – b/25 = 62/5

⇒ c = 0 and b = 10

∴ The value of b is 10.

The correct answer is Option 4.