Definition: A differential equation is an equation that relates a function to its derivatives; solving it means finding the function that makes the relationship true.
What is a differential equation?
It links an unknown function y with its derivative, for example dy/dx = 3y. A solution is a function, not a number. A general solution has a constant C; an initial condition pins down a particular solution.
How do you solve a separable one?
Separate the variables, then integrate both sides.
Worked example: dy/dx = 3y with y(0) = 5. Write dy/y = 3 dx. Integrate: ln|y| = 3x + C, so y = Ae3x. The condition y(0) = 5 gives A = 5. The solution is y = 5e3x.
Check: y’ = 15e3x = 3y.
This model describes populations that grow at a rate proportional to their size.
What goes wrong?
Forgetting the constant of integration before using the initial condition, which changes the answer. Mathbot adds C right at the integration step.