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00:00 - 00:59 | hello friends welcome to doubtnut question is solve the differential equation which is X into x square minus 1 into DY by DX is equal to one given that when x is equal to 2 Y is equal to zero not that we have given a differential equation which is X time of x square - 1 DY by DX is equal to 1 I can shift this X time of x square - 1 towards the denominator of RHS so the term is become a DY by DX is equal to 1 upon X time of x square minus one I can write this one upon X time of x minus 1 into X + b cos x square minus 1 square can be written as x minus 1 into X + 1 after that I can ship this DX towards the other side so the term is become a D Y is equal to |

01:00 - 01:59 | one upon X time of x minus 1 into X + 1 then after integrate both side now integrate both side integrate both side what we will get we get integration of D Y is equal to integration of 1 upon X into x minus 1 into X + 1 DX ok I can solve this integration by using a partial fraction ok so first we personalize this term then after integrate so I can put this is our equation number ok so first I parcelize disturb so I can consider it so I can consider light light map of x is equal to 1 upon X time of x minus 1 into X + 1 we know that by using partial fraction this term is written as |

02:00 - 02:59 | upon X + b upon x minus 1 + 3 upon X + 1 this is our equation number second after that I can use it doing LCM LCM s x x - 1 X + 1 number comply with a with x minus 1 into X + 1 + b x time of X + 1 + X time of x minus 1 is equal to 1 upon X time of x minus 1 into X + 1 this bottamma cancer and this equation become a 1 is equal to a time of x minus 1 into X + 1 + b time be time of X time of X + Y + see time of X time of x minus 1 this is our equation number 3rd ok I can put x is equal to zero in equation number |

03:00 - 03:59 | what we get 1 is equal to this term and this item becomes zero because if 0 x with any number is zero and disturb become a time of 0 -1 in 20 plus one that makes -12 the value of a is -1 then I can put this x minus 1 is equal to zero that makes x is equal to 1 if x is equal to 1 then this term and disturb both are zero only 20 remaining so 1 is equal to be time of 1 into 1 + 1 is to determine the value of B is 1 by 2 and third is I can put X + 1 is equal to zero that Mr x is equal to -1 if I can put x is equal to -1 then A and B both are zero onley C is remaining so 1 is equal to see time of -1 time of -1 -1 become -2 so the value of C |

04:00 - 04:59 | is value of C is one upon the value of C is also one upon two now after that I can put the value of A B and C in equation number S and we know that X time of x minus 1 into X + 1 is written as minus one upon X + 1 by 2 upon x minus 1 + 1 by 2 upon X + 1 and substitute this whole term in the equation number for server integration is become integration is become written as integration of DY is equal to integration of minus one upon X + 1 by 2 X + 1 + 1 upon 2 x minus 1 then ok so the integration of device become a y and I can separate this integral sweet become one upon x dx + 1 upon |

05:00 - 05:59 | 2 time of integration of 1 upon X + 1 + 1 upon 2 time of integration of 1 upon x minus one and we know that the integration of 1 upon X is Allen of Mod of X + integration of 1 upon X + 1 is a line of Mod of X + 1 and integration of 1 upon x minus 1 is a line of Mod of x minus 1 and + c is our integration constant this is our equation number 4 and this is our general solution of given differential equation but we have given that the initial condition we have given that we have given that x is equal to 2 by a given that Y is equal to zero when x is equal to 25 can put Y is equal to zero and value of access to the Mod of 2 is become a 2 + 1 by 2 Alan of |

06:00 - 06:59 | mode of 2 + 1 is 38 - 3 + 1 by 2 Ellen of mode of 2 - 1 is becoming one plus C we know that a line of 1 0 9 of 10 then the value of C is value of C is Allen of 2 minus 1 by 2 3 ok this is after that I can substitute the value of c in equation number 4th ok question number 4 also I can written as a line of 2 - 3 to the power 1 by 2 ok bye the help of property log m to the power n is written as and lock app ok and I can apply also one more property which is log M upon an is written as log M - login ok so I can write this Allen of 2 upon |

07:00 - 07:59 | square root of this is the value of C I can put the value of sin equation number 4 so we get the final solution of given differential equation ok show the value of differential equation is - time of Allen of Mod of X + 1 by 2 time of Allen of Mod of X + 1 + 1 by 2 time of Allen of Mod of x minus 1 + 2 upon under root 3 this is our final solution of given differential equation hope you understand this question thank you for signing doubtnut |

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