Explanation Given: x ′ 1 = 3 x 1 + 2 x 2 , x ′ 2 = − x 1 + 4 x 2 , x 1 ( 0 ) = − 1 , x 2 ( 0 ) = 1 . Approach: Take Laplace transformation of the given function and then take inverse Laplace transformation of the obtained function. L [ x ] = X ( s ) According to the First Shifting theorem, If L − 1 [ F ( s ) ] = f ( t ) , then L − 1 [ F ( s − a ) ] = e a t f ( t ) . Also, L − 1 [ s s 2 + b 2 ] = cos b t L − 1 [ b s 2 + b 2 ] = sin b t Calculation: Take Laplace transformation of the function x ′ 1 = 3 x 1 + 2 x 2 , L [ x ′ 1 ] − 3 L [ x 1 ] − 2 L [ x 2 ] = 0 s L [ x 1 ] − x 1 ( 0 ) − 3 L [ x 1 ] − 2 L [ x 2 ] = 0 s X 1 ( s ) − x 1 ( 0 ) − 3 X 1 ( s ) − 2 X 2 ( s ) = 0 s X 1 ( s ) − ( − 1 ) − 3 X 1 ( s ) − 2 X 2 ( s ) = 0 Solve further, ( s − 3 ) X 1 ( s ) − 2 X 2 ( s ) = − 1 … ( 1 ) Take Laplace transformation of the function x ′ 2 = − x 1 + 4 x 2 , L [ x ′ 2 ] + L [ x 1 ] − 4 L [ x 2 ] = 0 s L [ x 2 ] − x 2 ( 0 ) + L [ x 1 ] − 4 L [ x 2 ] = 0 s X 2 ( s ) − x 2 ( 0 ) + X 1 ( s ) − 4 X 2 ( s ) = 0 s X 2 ( s ) − 1 + X 1 ( s ) − 4 X 2 ( s ) = 0 Solve further, ( s − 4 ) X 2 ( s ) + X 1 ( s ) = 1 X 2 ( s ) = 1 − X 1 ( s ) s − 4 … ( 2 ) Substitute the value of X 2 ( s ) in equation ( 1 ) . ( s − 3 ) X 1 ( s ) − 2 ( 1 − X 1 ( s ) s − 4 ) = − 1 ( s − 3 ) ( s − 4 ) X 1 ( s ) − 2 + 2 X 1 ( s ) = − s + 4 X 1 ( s ) [ ( s − 3 ) ( s − 4 ) + 2 ] = − s + 6 X 1 ( s ) = − s − 6 s 2 − 7 s + 14 The value of L − 1 [ X 1 ( s ) ] will be, L − 1 [ X 1 ( s ) ] = L − 1 [ − s − 6 s 2 − 7 s + 14 ] x 1 ( t ) = − L − 1 [ s − 6 ( s − 7 2 ) 2 + ( 7 2 ) 2 ] = − L − 1 [ s − 7 2 ( s − 7 2 ) 2 + ( 7 2 ) 2 + − 5 2 ( s − 7 2 ) 2 + ( 7 2 ) 2 ] = − L − 1 [ s − 7 2 ( s − 7 2 ) 2 + ( 7 2 ) 2 + 5 7 − 7 2 ( s − 7 2 ) 2 + ( 7 2 ) 2 ] The value of L − 1 [ s s 2 + ( 7 2 ) 2 ] will be, L − 1 [ s s 2 + ( 7 2 ) 2 ] = cos ( 7 2 t ) = f ( t ) From the First Shifting theorem, the value of L − 1 [ s − 7 2 ( s − 7 2 ) 2 + ( 7 2 ) 2 ] will be,

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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The given initial-value problem.