Question

If a curve $y=f\left(x\right)$ passes through the point $(1,2)$ and satisfies $x\frac{dy}{dx}+y=b{x}^4,$ then for what value of $b,$ $\underset{1}{\overset{2}{\int }}f\left(x\right)dx=\frac{62}{5}$ ?

• A. $5$
• B. $\frac{62}{5}$
• C. $10$
• D. $\frac{31}{5}$

Answer 1

The correct option is C $10$

$\frac{dy}{dx}+\frac{y}{x}=b{x}^3$
I.F. $=e^{\int \frac{dx}{x}}=x$
$\therefore yx=\int b{x}^{4}dx=\frac{b{x}^5}{5}+c$
Above curve passes through $(1,2).$
$2=\frac{b}{5}+c\cdots \left(1\right)$

Also, $\underset{1}{\overset{2}{\int }}(\frac{b{x}^4}{5}+\frac{c}{x})dx=\frac{62}{5}$
$\Rightarrow \frac{b}{25}\times 32+c\ln 2-\frac{b}{25}=\frac{62}{5}\cdots \left(2\right)$

Solving $\left(1\right)$ and $\left(2\right),$
$c=0$ and $b=10$