Solve the following initial value problem: (1-t^2)y'-4ty =1, y(2)=-1 What is the largest interval on which its solution is guaranteed to uniquely exist?

Francisco Proctor 2022-07-25 Answered
Solve the following initial value problem:
( 1 t 2 ) y t 4 t y = 1 , y ( 2 ) = 1
What is the largest interval on which its solution is guaranteed to uniquely exist?
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Answers (1)

neobuzdanio
Answered 2022-07-26 Author has 13 answers
The homogeneous differential equation can be solved by separation of variables.
d y / y = 4 t d t / ( 1 t 2 ) is the separated differential equation.
Integration ln ( y ) = 2 ln ( 1 + t 2 ). Exponentiation: y ( t ) = a ( t 2 1 ) 2 . y ( 2 ) = a / 9 = 1. a = 9. y ( t ) = 9 / ( t 2 1 ) 2 .
The inhomogeneous differential equation is solved by: y ( t ) = ( 25 + 3 t t 2 ) / ( 3 ( 1 + t 2 ) 2 ).This is derived by variation of constants. y ( t ) = f ( t ) ( 1 + t 2 ) 2 . y ( t ) = f ( t ) ( 1 + t 2 ) 2 4 t f ( t ) ( 1 + t 2 ) 3 .So in the inhomogeneous differential equation: f ( t ) ( 1 + t 2 ) 1 + 4 t f ( t ) ( 1 + t 2 ) 2 4 t f ( t ) ( 1 + t 2 ) 2 = f ( t ) ( 1 + t 2 ) 1 = 1. Simply integrate and adopt the constant to rewrite the above result.
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