Question:
If a curve $y=f(x)$ passes through the point $(1,2)$ and satisfies $x \frac{\mathrm{d} y}{\mathrm{~d} x}+y=\mathrm{b} x^{4}$, then for what value of b, $\int_{1}^{2} f(x) \mathrm{d} x=\frac{62}{5} ?$
Correct Option: 4,
Solution:
$\frac{d y}{d x}+\frac{y}{x}=b x^{3} \cdot I . F .=e^{\int \frac{d x}{x}}=x$
$\therefore y x=\int b x^{4} d x=\frac{b x^{5}}{5}+c$
Passes through $(1,2)$, we get
$2=\frac{b}{5}+C \ldots \ldots(i)$
Also, $\int_{1}^{2}\left(\frac{b x^{4}}{5}+\frac{c}{x}\right) d x=\frac{62}{5}$
$\Rightarrow \frac{b}{25} \times 32+\operatorname{Cln} 2-\frac{b}{25}=\frac{62}{5} \Rightarrow C=0 \& b=10$
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