When working with differential equations in Julia, it is important to choose the right solver for the problem at hand. In this article, we will explore three different approaches to solving a simple example using a stiff ode solver in the DifferentialEquations.jl package.

## Approach 1: Using the `@ode_def` macro

The first approach involves defining the differential equation using the `@ode_def` macro. This macro allows us to specify the equation in a concise and readable manner. Here is the code:

```
using DifferentialEquations
@ode_def StiffODE begin
dy = -k*y
end
function solve_stiff_ode(k, y0, tspan)
prob = ODEProblem(StiffODE, y0, tspan)
sol = solve(prob, Tsit5())
return sol
end
k = 1.0
y0 = 1.0
tspan = (0.0, 1.0)
sol = solve_stiff_ode(k, y0, tspan)
```

In this approach, we define the differential equation using the `@ode_def` macro, which takes the form `@ode_def Name begin … end`. We then define a function `solve_stiff_ode` that takes the parameters `k`, `y0`, and `tspan`, and returns the solution to the differential equation. Finally, we call the `solve_stiff_ode` function with the desired parameters and store the solution in the variable `sol`.

## Approach 2: Using the `@ode` macro

The second approach involves using the `@ode` macro to define and solve the differential equation in a single step. Here is the code:

```
using DifferentialEquations
function solve_stiff_ode(k, y0, tspan)
@ode dy = -k*y
prob = ODEProblem(dy, y0, tspan)
sol = solve(prob, Tsit5())
return sol
end
k = 1.0
y0 = 1.0
tspan = (0.0, 1.0)
sol = solve_stiff_ode(k, y0, tspan)
```

In this approach, we define the differential equation using the `@ode` macro, which takes the form `@ode dy = …`. We then define a function `solve_stiff_ode` that takes the parameters `k`, `y0`, and `tspan`, and returns the solution to the differential equation. Inside the function, we use the `@ode` macro to define the differential equation, and then proceed to solve it and store the solution in the variable `sol`.

## Approach 3: Using the `@ode_def` macro with a custom solver

The third approach involves using the `@ode_def` macro to define the differential equation, and then solving it using a custom solver. Here is the code:

```
using DifferentialEquations
@ode_def StiffODE begin
dy = -k*y
end
function solve_stiff_ode(k, y0, tspan)
prob = ODEProblem(StiffODE, y0, tspan)
sol = solve(prob, MyCustomSolver())
return sol
end
k = 1.0
y0 = 1.0
tspan = (0.0, 1.0)
sol = solve_stiff_ode(k, y0, tspan)
```

In this approach, we define the differential equation using the `@ode_def` macro, just like in the first approach. We then define a function `solve_stiff_ode` that takes the parameters `k`, `y0`, and `tspan`, and returns the solution to the differential equation. Inside the function, we use the `@ode_def` macro to define the differential equation, and then solve it using a custom solver (represented by `MyCustomSolver()` in the code). The solution is stored in the variable `sol`.

After exploring these three approaches, it is clear that the second approach, using the `@ode` macro, is the most concise and readable. It allows us to define and solve the differential equation in a single step, making the code more compact and easier to understand. Therefore, the second approach is the recommended option for solving a simple example using a stiff ode solver in Julia.