Consider the equation y ' ' − y ' − 2 y = 0 (a). Show that y 1 ( t ) = e − t and y 2 ( t ) = e 2 t form a fundamental set of solutions. (b). Let y 3 ( t ) = − 2 e 2 t , y 4 ( t ) = y 1 ( t ) + 2 y 2 ( t ) , and y 5 ( t ) = 2 y 1 ( t ) − 2 y 3 ( t ) Are y 3 ( t ) , y 4 ( t ) and y 5 ( t ) also solution of the given differential equation? (c). Determine whether each of the following pair forms a fundamental set of solution: [ y 1 ( t ) , y 3 ( t ) ] ; [ y 2 ( t ) , y 3 ( t ) ] ; [ y 1 ( t ) , y 4 ( t ) ] ; [ y 4 ( t ) , y 5 ( t ) ] .
Consider the equation y ' ' − y ' − 2 y = 0 (a). Show that y 1 ( t ) = e − t and y 2 ( t ) = e 2 t form a fundamental set of solutions. (b). Let y 3 ( t ) = − 2 e 2 t , y 4 ( t ) = y 1 ( t ) + 2 y 2 ( t ) , and y 5 ( t ) = 2 y 1 ( t ) − 2 y 3 ( t ) Are y 3 ( t ) , y 4 ( t ) and y 5 ( t ) also solution of the given differential equation? (c). Determine whether each of the following pair forms a fundamental set of solution: [ y 1 ( t ) , y 3 ( t ) ] ; [ y 2 ( t ) , y 3 ( t ) ] ; [ y 1 ( t ) , y 4 ( t ) ] ; [ y 4 ( t ) , y 5 ( t ) ] .
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