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Q.
If a curve $y=f(x)$ passes through the poin $(1,2)$ and satisfies $x \frac{d y}{d x}+y=b x^{4}$, then for what value of $b, \int_{1}^{2} f(x) d x=\frac{62}{5} ?$
$\frac{d y}{d x}+\frac{y}{x}=b x^{3}$
I.F. $=e^{\frac{1}{ x } dx }= x$
So, solution of D.E. is given by
$y \cdot x=\int b \cdot x^{3} \cdot x d x+c$
$y=\frac{c}{x}+\frac{b x^{4}}{5}$
Passes through (1,2)
$2=c+\frac{b}{5}$...(1)
$\int_{1}^{2} f(x) d x=\frac{62}{5}$
$\left[ c \ln x +\frac{ bx ^{5}}{25}\right]_{1}^{2}=\frac{62}{5}$
$c \ln 2+\frac{31 b }{25}=\frac{62}{5}$...(2)
By equation (1)$\&(2)$
$c=0$ and $b=10$