Introduction to Differential Equations
Understand what differential equations are, learn to solve separable equations, and distinguish between general and particular solutions.
What Is a Differential Equation?
A differential equation is an equation that involves an unknown function and one or more of its derivatives. In Year 12, we focus on first-order ordinary differential equations (ODEs) involving dy/dx.
For example, dy/dx = 3x is a differential equation. Solving it means finding a function y = f(x) whose derivative equals 3x.
Examples of Differential Equations
Separable Differential Equations
A differential equation is separable if it can be written in the form dy/dx = g(x) . h(y), where the right-hand side is a product of a function of x alone and a function of y alone.
To solve, we separate the variables:
- Rearrange so all y terms are on one side and all x terms on the other: (1/h(y)) dy = g(x) dx
- Integrate both sides: ∫ (1/h(y)) dy = ∫ g(x) dx
- Add the constant of integration C
- If possible, solve for y explicitly
Important: Only one constant of integration is needed. When integrating both sides, we combine the two constants into a single C.
General and Particular Solutions
The general solution of a differential equation contains an arbitrary constant C. It represents an entire family of curves. A particular solution is found by using an initial condition (a known point) to determine the value of C.
Visual: Family of Solution Curves
General Solution
dy/dx = 2x gives y = x^2 + C
Infinitely many parabolas, shifted vertically
Particular Solution
If y(0) = 3, then 3 = 0 + C, so C = 3
One specific curve passing through (0, 3)
Key Vocabulary
Differential Equation
An equation relating a function to its derivatives. We seek the unknown function.
Separable
A differential equation where the variables x and y can be placed on opposite sides of the equation.
General Solution
The solution containing an arbitrary constant C, representing a family of curves.
Initial Condition
A known value of the function (e.g., y(0) = 5) used to find the particular solution.
Worked Examples
Solve dy/dx = 3x^2, given y(1) = 5.
Step 1: Integrate both sides with respect to x: y = ∫ 3x^2 dx = x^3 + C
Step 2: Apply the initial condition y(1) = 5: 5 = 1^3 + C, so C = 4
Answer: y = x^3 + 4
Solve dy/dx = y, given y(0) = 2.
Step 1: Separate variables: (1/y) dy = dx
Step 2: Integrate both sides: ln|y| = x + C
Step 3: Solve for y: y = Ae^x where A = e^C
Step 4: Apply initial condition: 2 = Ae^0 = A, so A = 2
Answer: y = 2e^x
Find the general solution of dy/dx = x/y.
Step 1: Separate variables: y dy = x dx
Step 2: Integrate both sides: y^2/2 = x^2/2 + C
Step 3: Simplify: y^2 = x^2 + K (where K = 2C)
Answer: y^2 - x^2 = K (a family of hyperbolas)
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the general solution of dy/dx = 4x?
Question 2
Which of the following is a separable differential equation?
Question 3
The general solution of dy/dx = y is y = Ae^x. If y(0) = 7, what is the particular solution?
Question 4
To solve dy/dx = x/y by separation of variables, the first step is:
Question 5
What distinguishes a particular solution from a general solution?
Key Concepts Summary
- ● A differential equation involves an unknown function and its derivatives.
- ● Separable equations can be written as g(x) dx = h(y) dy and solved by integrating both sides.
- ● The general solution contains a constant C; the particular solution uses an initial condition to find C.
- ● Always check your solution by substituting back into the original equation.
- ● Remember to include absolute value signs when integrating 1/y, and use A = ±e^C to handle the sign.