^{1}

^{2}

^{1}

^{2}

We will consider a continuously differentiable function

The question concerning the stability of functional equations has been originally raised by Ulam [

If the answer to this question is affirmative, the functional equation

Usually the experiment (or the observed) data do not exactly coincide with theoretical ones. We may express natural phenomena by use of equations but because of the errors due to measurement or observance the actual experiment data can almost always be a little bit off the expectations. If we would use inequalities instead of equalities to explain natural phenomena, then these errors could be absorbed into the solutions of inequalities; that is, those errors would be no more errors.

There is another way to explain the Hyers-Ulam stability. Let us consider a closed system which can be explained by the first-order linear differential equation, namely,

Even though the system is not predictable exactly because of outside disturbances, we say the differential equation

A generalization of Ulam’s problem was recently proposed by replacing functional equations with differential equations. Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [

The above result by Alsina and Ger was generalized by Miura et al. [

Miura et al. [

Wang et al. [

Let

We wondered if a stability of (

In this paper, we are going to prove the Hyers-Ulam stability of the differential equation (

Throughout this paper, let

In the following theorem, we shall prove the Hyers-Ulam stability of the differential equation (

Let

Assume that

Integrating each term of (

It follows from (

If we define a function

By an argument similar to the above, for the case when

Let

In view of (ii), (iii), and (iv), it is not difficult to show that

In the next theorem, we are going to generalize the stability result of the differential equation (

Let

Similarly to Theorem

Let

As in the proof of Corollary

In this section, we also assume that

Let

Assume that

It follows from (

It obviously follows from (ii), (iii), and (iv) that

By an argument similar to the above, for the case when

We remark that if

For any

Let

In this paper, we proved the Hyers-Ulam stability of the linear differential equation

We now define a subinterval

This relation implies that if

The authors declare that they have no competing interests.

All authors read and approved the final paper.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2015R1D1A1A01059467) and Hallym University Research Fund, 2014 (HRF-201409-017). Soon-Mo Jung was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2015R1D1A1A02061826).