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If a curve y=f\left ( x \right ) passes through the point \left ( 1,2 \right ) and satisfies x\frac{dy}{dx}+y=bx^{4}, then for what value of b, \int_{1}^{2}f\left ( x \right )dx=\frac{62}{5} ?
 
Option: 1 \frac{31}{5}
Option: 2 \frac{62}{5}
Option: 3 5
Option: 4 10

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\begin{aligned} &\frac{d y}{d x}+\frac{y}{x}=b x^{3}\\ &\text { I.F. }=\mathrm{e}^{\frac{1}{\mathrm{x}} \mathrm{dx}}=\mathrm{x}\\ &\text { So, solution of D.E. is given by } \end{aligned}

\begin{aligned} &y \cdot x=\int b \cdot x^{3} \cdot x d x+c\\ &y=\frac{c}{x}+\frac{b x^{4}}{5}\\ &\text {Passes through }(1,2) \end{aligned}

2=c+\frac{b}{5}\qquad\ldots(1)

\\\int_{1}^{2} f(x) d x=\frac{62}{5} \\\\ {\left[\operatorname{c} \ln x+\frac{b x^{5}}{25}\right]_{1}^{2}=\frac{62}{5}}

\begin{aligned} &\mathrm{c} \ln 2+\frac{31 \mathrm{~b}}{25}=\frac{62}{5}\\ &\text {By equation (1) \& (2) }\\ &\mathrm{c}=0 \text { and } \mathrm{b}=10 \end{aligned}

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Suraj Bhandari

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