Prove that there are infinitely many different solutions of the differential equations

There would be infinite solutions if no initial-value problem is specified. For example, take the differential equation written below:

dy = 3dx

When you differentiate that, the equation would become:

y - y₀ = 3 (x - x₀)

Now, there can be arbitrary values of x₀ and y₀. If no initial-value problem is specified, you cannot solve the problem because there are infinite solution. Example of an initial value problem is: when x₀ = 2, y₀ = 2. If we had that, we can find a solution to the differential equation.