AcademyDifferential Equations
Academy
First-Order Differential Equations
Level 1 - Calculus topic page in Differential Equations.
Principle
A first-order differential equation relates an unknown function to its first derivative. Solving it means finding functions that satisfy the derivative relationship.
Notation
\(y'\)
first derivative of y with respect to x
\(\frac{dy}{dx}\)
same first derivative in differential notation
\(C\)
constant determined by an initial condition
The Core Method
A common linear form is
First-order linear equation
\[y'+p(x)y=q(x)\]
For direct equations such as \(dy/dx=f(x)\), integrate both sides. For separable equations, separate variables before integrating.
Worked Cases
Question
Solve \(dy/dx=3x^2\).
Answer
Integrate both sides with respect to \(x\): \(y=\int3x^2dx=x^3+C\). Every function \(y=x^3+C\) has derivative \(3x^2\).
Examples
Question
Solve \(y'=2x\) with \(y(1)=5\).
Answer
Integrate: \(y=x^2+C\). Use \(y(1)=5\): \(1+C=5\), so \(C=4\). Therefore \(y=x^2+4\).