EGM4313.HW9 - 1 Let be a fundamental matrix for the problem...

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1) Let be a fundamental matrix for the problem y Ay that has the property that (0) . I Show that 1 ( ) ( ) Z t s and 2 () Z t s both satisfy the initial value problem , (0, ) ( ) Z AZ Z s s . Solution: Since t is a fundamental matrix for y Ay we have ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( )) ( ) ( ). t A t t s A t s t s A t s Therefore ( ) ( ) ts satisfies the equation. We also have (0) ( ) ( ) ( ) s I s s For the initial condition. Since t is a fundamental matrix for y Ay we have ( ). d A d Let ( ). d t s t s A t s dt Therefore satisfies the equation. We also have 0 ( ) ( ) t t s s so exactly the same initial condition is satisfied. Since linear initial value problems have unique solutions and we find that and ( ) ( ) satisfy the same initial value problem we can conclude that they are equal. 2) Find the general solution to t t e e y y 2 2 4 1 1 2 by the method of variation of parameters as described in our text and by the method of variation of parameters using the special fundamental matrix as described in class. Solution: A fundamental matrix for the homogeneous equations is easily found and is t t t t e e e e Y 2 3
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EGM4313.HW9 - 1 Let be a fundamental matrix for the problem...

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