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Differential equations in physics
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Most laws are equations linking a quantity to its rates of change.
d²x/dt² = −ω²xMany laws say how a quantity changes in terms of itself, such as d²x/dt² = −ω²x. Solving means finding functions that satisfy the equation, then using initial conditions to fix constants. Exponentials describe growth and decay and sinusoids describe oscillation.
Radioactive decay dN/dt = −λN gives N = N₀e^{−λt}.
The solution is a family; you need initial or boundary conditions to choose one.
In more detail
A differential equation relates a function to its derivatives. Newton's second law, mass times acceleration equals force, is a second-order differential equation for position.
Real-world uses
- Predicting the motion of a spring, pendulum or planet uses differential equations.
- Circuit analysis with capacitors and inductors leads to differential equations solved to find current over time.
- Engineers use the equation of motion for a spring-mass system to predict how a car suspension responds to bumps in the road.
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Frequently asked questions
Why are differential equations central to physics?
They describe how systems change from one moment to the next, which is the basis of prediction.
Do I need to solve them by hand?
In introductory courses you solve simple ones by hand. Computers handle complex ones numerically.
Are there simple cases I can solve by hand?
Yes. Constant acceleration and simple harmonic motion have standard solutions that you can check by substitution.
Related topics in Measurement & Math Tools
Calculus: integrals · Partial derivatives · Gradient, divergence, curl · Taylor series and small angles · Complex numbers and phasors · Graphs and slopes