Question

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

Answer 1

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.