Solve this following


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 $b, \int_{1}^{2} f(x) d x=\frac{62}{5} ?$


  1. 5

  2. 10

  3. $\frac{62}{5}$

  4. $\frac{31}{5}$

Correct Option: 2,



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 $\mathrm{b}=10$


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