Explanation The given system of equation is, t x ′ = ( 2 − 1 3 − 2 ) x + ( 1 − t 2 2 t ) … ( 1 ) The given vector is, x ( c ) = c 1 ( 1 1 ) t + c 2 ( 1 3 ) t − 1 . Consider the homogeneous part of the given equation. That is, t x ′ = ( 2 − 1 3 − 2 ) x … ( 2 ) Substitute the given x ( c ) in the above equation (2). ( c 1 ( 1 1 ) t + c 2 ( 1 3 ) t − 1 ) x ′ = c 1 ( 2 − 1 3 − 2 ) ( 1 1 ) t + c 1 ( 2 − 1 3 − 2 ) ( 1 3 ) t − 1 … ( 2 ) = c 1 ( 1 1 ) t + c 2 ( 1 3 ) t − 1 Thus, x ( c ) is a solution. The fundamental matrix is, Ψ ( t ) = ( t t − 1 t 3 t − 1 ) The inverse of the fundamental matrix is obtained as, Ψ − 1 ( t ) = 1 2 ( 3 t − t − 1 − t t ) Use the method of variation of parameters to obtain the solution of the IVP. It is noted from the given equation that, g ( t ) = ( 1 − t 2 2 t )

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ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 7.9, Problem 14P

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To verify: The given vector, x(c)=c1(11)t+c2(13)t−1 is the general solution of the homogeneous system of tx′=(2−13−2)x+(1−t22t) and hence solve the non homogeneous system (t>0).

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Chapter 7.9, Problem 13P

Chapter 7.9, Problem 15P

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To verify: The given vector , x ( c ) = c 1 ( 1 1 ) t + c 2 ( 1 3 ) t − 1 is the general solution of the homogeneous system of t x ′ = ( 2 − 1 3 − 2 ) x + ( 1 − t 2 2 t ) and hence solve the non homogeneous system ( t &gt; 0 ) .

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To verify: The given vector, x(c)=c1(11)t+c2(13)t−1 is the general solution of the homogeneous system of tx′=(2−13−2)x+(1−t22t) and hence solve the non homogeneous system (t>0).

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To verify: The given vector , x ( c ) = c 1 ( 1 1 ) t + c 2 ( 1 3 ) t − 1 is the general solution of the homogeneous system of t x ′ = ( 2 − 1 3 − 2 ) x + ( 1 − t 2 2 t ) and hence solve the non homogeneous system ( t > 0 ) .