A differential equation relates a function to its own derivative. The unknown is a whole function.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Separating
Gather every y with dy and every x with dx, then integrate both sides.
The constant matters
Integrating introduces + C. Without it you have one solution instead of the family.
Pinning it down
An initial condition picks the one member of the family that passes through a given point.
An equation whose unknown is a function
A differential equation relates a function to its derivatives. Solving it means finding the function, not a number, and the answer is generally a family of functions rather than one.
Separating the variables
Rearrange so all the y terms sit with dy and all the x terms with dx, then integrate both sides. The method applies only when the equation separates, which is why it is checked first.
The constant and the initial condition
Integrating introduces a constant, giving a family of solutions. An initial condition picks out one member. The exam usually supplies one and expects the particular solution, not the general one.
State the domain of the solution
A particular solution is valid on the interval containing the initial condition where it stays defined. Reporting a solution without noting where it applies is incomplete, and the exam has asked for it.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Nadia Solve One
Nadia solves dy/dx = 2x with y = 5 when x = 0.
- Step 1
The variables are already separated: dy = 2x dx.
Step 4: Your Turn
Practice makes it stick.
The Constant
Problem 1 of 2
dy/dx = 2x with y = 7 at x = 0. What is C?
Evaluating
Problem 2 of 2
With y = x² + 7, what is y at x = 3?
Separate and Integrate
1 of 8
dy/dx = 4x with y = 1 at x = 0. What is C?
2 of 8
dy/dx = 3 with y = 2 at x = 0. What is y at x = 4?
3 of 8
Does a differential equation have one solution or a family? 1 for one, 2 for a family.
4 of 8
Order the steps for solving a separable equation with an initial condition.
- 1Integrate both sides
- 2Add the constant of integration
- 3Apply the initial condition
- 4Separate the variables
5 of 8
dy/dx = 2x with y = 0 at x = 2. What is C?
6 of 8
Exponential growth solves dy/dx = ky. Is the rate proportional to the amount? 1 for yes, 0 for no.
7 of 8
dy/dx = 6x² with y = 1 at x = 0. What is y at x = 1?
8 of 8
Forgetting + C loses how many solutions? 1 for one, 2 for infinitely many.
Step 5: Quick Check
Show what you know.
Question 1 of 1
dy/dx = 2x with y = 9 at x = 0. What is y at x = 2?
What You Learned
- A differential equation relates a function to its derivative.
- Separate the variables, integrate both sides, and keep the constant.
- An initial condition selects one member from the family of solutions.