A differential equation relates a function to its derivatives. Ordinary differential equations (ODEs) involve one independent variable, usually time. They are the native language of growth, decay, oscillators, and many engineering models.
What a solution means
Solving $y'=f(t,y)$ means finding a function $y(t)$ whose slope at each point matches the rule $f$. An initial condition $y(t_0)=y_0$ selects one solution curve from a family. Existence and uniqueness theorems tell you when that selection is reliable — roughly, when $f$ is sufficiently well-behaved near the starting point.
Separable equations
If $y'=g(t)h(y)$, you can often rearrange and integrate both sides. That technique is a workhorse for first courses. Walk through it in detail in Separable differential equations walkthrough.
Qualitative pictures
Before solving, sketch a slope field: at each point $(t,y)$, draw a short segment with slope $f(t,y)$. Solution curves must thread those segments. Equilibria where $y'=0$ and whether nearby solutions approach or flee them often matter more than a closed-form formula.