If , and are solutions to the initial-value problem , with , then for all t.

Question

If  , and  are solutions to the initial-value problem  , with  , then  for all t.

Quizplus · Accepted Answer

Since both 11edaba5_20a7_baf1_a975_ef3d090bd40b_TBMC1039_00 and 11edaba5_20a7_baf2_a975_23ee960fc314_TBMC1039_00 are solutions to the same initial-value problem, they must satisfy the same differential equation and initial condition. Therefore, their difference, 11edaba5_20a7_baf2_a975_23ee960fc314_TBMC1039_00 - 11edaba5_20a7_baf1_a975_ef3d090bd40b_TBMC1039_00, must satisfy the homogeneous version of the differential equation with the same initial condition. But this difference is simply a constant, so it cannot depend on t. Therefore, the solution 11edaba5_20a7_baf5_a975_4b2795c896b7_TBMC1039_00, which is obtained by adding this constant to either 11edaba5_20a7_baf1_a975_ef3d090bd40b_TBMC1039_00 or 11edaba5_20a7_baf2_a975_23ee960fc314_TBMC1039_00, will also satisfy the differential equation and initial condition.

If , and Are Solutions to the Initial-Value