Question

If a curve $y=f(x)$ passes through the poin $(1,2)$ and satisfies $x\frac{dy}{dx}+y=b{x}^4$, then for what value of $b,{\int }_1^{2}f(x)dx=\frac{62}{5}?$

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

Answer 1

Answer: (B)

$\frac{dy}{dx}+\frac{y}{x}=b{x}^3$
I.F. $=e^{\frac{1}{x}dx}=x$
So, solution of D.E. is given by
$y\cdot x=\int b\cdot x^3\cdot xdx+c$
$y=\frac{c}{x}+\frac{b{x}^4}{5}$
Passes through (1,2)
$2=c+\frac{b}{5}$...(1)
${\int }_1^{2}f(x)dx=\frac{62}{5}$
${\left[c\ln x+\frac{b{x}^5}{25}\right]}_1^2=\frac{62}{5}$
$c\ln 2+\frac{31b}{25}=\frac{62}{5}$...(2)
By equation (1)$\&(2)$
$c=0$ and $b=10$