Question:
If a curve $y=f(x)$ passes through the point
$(1,2)$ and satisfies $x \frac{d y}{d x}+y=b x^{4}$, then for what
value of $b, \int_{1}^{2} f(x) d x=\frac{62}{5} ?$
Correct Option: 2,
Solution:
$\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{\mathrm{y}}{\mathrm{x}}=\mathrm{bx}^{3}$
I.F. $=e^{\frac{1}{x} d x}=x$
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 $(1,2)$
$2=\mathrm{c}+\frac{\mathrm{b}}{5}$ ..............(1)
$\int_{1}^{2} f(x) d x=\frac{62}{5}$
$\left[\mathrm{c} \ln \mathrm{x}+\frac{\mathrm{bx}^{5}}{25}\right]_{1}^{2}=\frac{62}{5}$
$\operatorname{cln} 2+\frac{31 \mathrm{~b}}{25}=\frac{62}{5}$ ...........(2)
By equation (1) \& (2)
$\mathrm{c}=0$ and $\mathrm{b}=10$