EC-3.1 First-Order Equations, and the Circuits That Obey Them
The standard first-order treatment as taught in engineering mathematics courses, September 2026
What this is and why it exists
An ordinary equation asks you for a number. A differential equation asks you for a function, and it asks by telling you about that function's rate of change.
That sounds abstract until you notice how much of engineering arrives in exactly this form. A capacitor charging, a body cooling, a tank filling, a population growing: all of them are described by one sentence. Something changes at a rate proportional to how far it is from where it is heading.
So a first-order equation is not one topic among many. It is the shape most physical situations take the first time you write them down. Being fluent with it means you can predict a circuit's behaviour before you solve anything.
The vocabulary
- Order — the highest derivative that appears. First order means the first derivative appears and nothing higher.
- Linear — the unknown function and its derivatives appear only to the first power, never multiplied together, never inside another function.
- Separable — every appearance of the unknown can be moved to one side and every appearance of the independent variable to the other.
- General solution — the whole family of functions that satisfy the equation. It carries one arbitrary constant for a first-order equation.
- Particular solution — the one member of that family picked out by an initial condition.
- Initial condition — the value of the unknown at some starting instant.
- Integrating factor — a function you multiply through by so that one side becomes the derivative of a product.
- Time constant — the time a first-order response takes to cover about sixty-three percent of the distance to its final value.
The mental model
Start with the equation that matters most. The derivative of some quantity with respect to time equals minus that quantity, divided by a constant called tau.
Read it as a sentence. The rate of change is proportional to the value, and the minus sign says it moves toward zero. When the value is large the change is fast, and as the value shrinks the change slows. Nothing ever quite arrives, which is why the answer is an exponential and not a straight line.
The solution is the starting value multiplied by e raised to the power minus t over tau. You should be able to write that down without deriving it, because it is the single most common answer in the subject.
Tau is the whole story, and it is visible before you solve. It is one divided by the coefficient sitting in front of the unknown. In a resistor and capacitor in series it comes out as resistance multiplied by capacitance. In a resistor and inductor it is inductance divided by resistance. After one tau the response has covered about sixty-three percent of its journey. After three tau it is at about ninety-five percent. After five tau it is finished for any practical purpose.
Now the two methods.
Separation works when the two variables can be pulled apart. Put everything with the unknown on one side and everything with time on the other. Integrate both sides once and you are done except for the constant. Most textbook growth and decay problems are this shape wearing different words.
The integrating factor works for any linear first-order equation, including the ones separation cannot touch. The equation has the shape: the derivative of y plus some function of t multiplied by y equals some other function of t. Multiply the whole equation by e raised to the integral of that first function. The left side then becomes exactly the derivative of a product, so one integration finishes it. It is worth deriving that factor once so it stops looking like a rule from nowhere.
Last, the step people skip. The general solution is a family of curves differing by one constant. Until you apply the initial condition you have a shape, not an answer. A solution that is correct in form and wrong in value is still wrong.
What you should now be able to explain or do
- Say what makes an equation first order and what makes it linear, given one to look at.
- Solve a separable equation and a linear one, and choose the right method by inspection.
- Read the time constant off an RC or RL circuit without solving the equation.
- Explain why the response is exponential rather than linear, in terms of the rate depending on the value.
- Apply an initial condition and say what would be wrong with an answer that had not.
Check yourself
A circuit has a one kilohm resistor and a ten microfarad capacitor in series. What is the time constant, and roughly how long until the capacitor is fully charged for practical purposes?
The time constant is resistance times capacitance, so ten milliseconds. Five time constants is the usual practical rule, so about fifty milliseconds.
Why does a first-order response approach its final value without ever reaching it?
Because the rate of change is proportional to the distance remaining. As the distance shrinks the rate shrinks with it, so the approach slows without ever stopping.
You have solved an equation and obtained a general solution with an arbitrary constant in it. What is still missing?
The initial condition. It selects the one member of the family that matches the physical situation at the starting instant.
When would you reach for an integrating factor rather than separation?
When the equation is linear but not separable, which happens as soon as there is a forcing term that depends on time. Separation cannot pull the variables apart in that case.
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