EC-3.1 First-Order Equations, and the Circuits That Obey Them
You can recognise a first-order equation, solve it by separation or by an integrating factor, and read the time constant of an RC or RL circuit straight off the equation that describes it.
Before:nothing requiredUnlocks:EC-4. Transforms, Complex Variables and Numerical Methods
A differential equation is a statement about a rate. Almost every first-order equation an engineer meets says the same thing — something changes at a rate proportional to how far it is from where it is heading — and that one sentence produces the exponential settling seen in a charging capacitor, a cooling body and a filling tank. The skill worth building here is not the integration but the reading: seeing the time constant in the coefficient before solving anything, and knowing that a general solution is a family of curves until an initial condition picks one out.
Work through these
What a differential equation states, and what counts as a solution
An algebraic equation asks for a number and a differential equation asks for a function, so a solution is checked by substituting it back and seeing the statement hold everywhere. Order is the highest derivative present, and it decides how many arbitrary constants the general solution carries.
NPTEL: Differential Equations for Engineers · CourseSeparable equations, and the single integration that finishes them
When every appearance of the unknown can be moved to one side and every appearance of the variable to the other, integrating both sides once is the whole method. Most textbook growth and decay problems are this shape wearing different words.
NPTEL: Differential Equations for Engineers · CourseThe linear first-order form, and the integrating factor that makes it exact
An equation linear in the unknown and its first derivative is solved by multiplying through by a factor chosen so the left side becomes the derivative of a product. The factor is an exponential of an integral of the coefficient, and it is worth deriving once rather than memorising.
MIT OpenCourseWare 18.03: Differential Equations · CourseThe time constant, and what it means before you solve anything
The coefficient in front of the unknown carries the time constant, which says how long the response takes to cover most of the distance to its final value. Reading it off the equation is faster than solving and is often the only number a design question needs.
Initial conditions, and why a general solution is not yet an answer
The general solution is a family of curves differing by one constant, and the initial condition selects the member that matches the physical situation at the starting instant. Forgetting this step gives an answer that is correct in form and wrong in value.
Solve one RC circuit twice: from its equation, and from the physics
Writing the current law for a resistor and capacitor gives a first-order equation whose solution is the familiar exponential charging curve. Reaching the same curve by reasoning about the capacitor's stored charge is the check that the mathematics and the circuit are the same statement.
NPTEL: Differential Equations for Engineers · Course
Sign in to keep your progress.
Free resources
We haven't checked most of these for screen reader use yet.
Links last checked 3 Sept 2026.
Stuck here?
Ask a mentor. A real person answers, and they can see exactly which topic you're on. Usually within a couple of working days.
Checking your session…
Topics shown in module order.