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Mathematics Question on Differential equations

dydx\frac{dy}{dx} + 5x(1+x5)\frac{5}{x(1+x^5)}y = (1+x5)2x7\frac{(1+x^5)^2}{x^7} If y(1) = 2, then the value of y(2) is:

A

693128\frac{693}{128}

B

697128\frac{697}{128}

C

637128\frac{637}{128}

D

627128\frac{627}{128}

Answer

693128\frac{693}{128}

Explanation

Solution

The correct option is (A): 693128\frac{693}{128}
I.F=e∫5x(1+x5)dx=e∫5x−6(x−5+1)dxI.F=e^{\int\frac{5}{x(1+x^5)}dx}=e^{\int\frac{5x^{-6}}{(x^{-5}+1)}dx}
=e−ln(x−5+1)=1x−5+1=x5x5+1=e^{-ln{}(x^{-5}+1)}=\frac{1}{x^{-5}+1}=\frac{x^5}{x^5+1}
y.x5x5+1=∫(1+x5)2x7.x5(1+x5)dxy.\frac{x^5}{x^5+1}=\int\frac{(1+x^5)^2}{x^7}.\frac{x^5}{(1+x^5)}dx
=∫(1+x5)x2dx=\int\frac{(1+x^5)}{x^2}dx
=−1x+x44+C=\frac{-1}{x}+\frac{x^4}{4}+C
y(1)=2⇒2(12)=−1+14+Cy(1)=2⇒2(\frac{1}{2})=-1+\frac{1}{4}+C
⇒C=74⇒C=\frac{7}{4}
Put x=2
⇒y(3233)=−12+4+74⇒y(\frac{32}{33})=\frac{-1}{2}+4+\frac{7}{4}
⇒y=693128⇒ y=\frac{693}{128}