Variable separable differential equations: Class 12 practice
If dy/dx = f(x)g(y), separate: dy/g(y) = f(x) dx, integrate both sides and add a constant. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Integration
The idea behind variable separable differential equations
If dy/dx = f(x)g(y), separate: dy/g(y) = f(x) dx, integrate both sides and add a constant. Use an initial condition to find it.
Step-by-step method
- Move all y terms with dy and x terms with dx.
- Integrate both sides.
- Add + C, then use the initial condition.
Worked example 1 (Basic)
Solve dy/dx = 4x given y(0) = 3.
- y = 2x² + C
- C = 3
- Answer: y = 2x² + 3
Worked example 2 (Standard)
Solve dy/dx = 2y given y(0) = 4.
- dy/y = 2 dx → ln y = 2x + C
- y = 4e^(2x)
- Answer: y = 4e^(2x)
Worked example 3 (Challenge)
Solve dy/dx = x/y given y(0) = 2 (y > 0).
- y dy = x dx → y²/2 = x²/2 + C
- y² = x² + 4
- Answer: y = √(x² + 4)
Common mistakes and misconceptions
- Forgetting the constant
- Integrating before separating
Practice questions
- (Basic) Solve dy/dx = 6x given y(0) = 3.
- (Basic) Solve dy/dx = 4x given y(0) = 1.
- (Standard) Solve dy/dx = 3y given y(0) = 3.
- (Standard) Solve dy/dx = 1y given y(0) = 3.
- (Challenge) Solve dy/dx = x/y given y(0) = 3 (y > 0).
- (Challenge) Solve dy/dx = x/y given y(0) = 1 (y > 0).
Answer key with full working
- Q1: y = 3x² + C → C = 3 → Answer: y = 3x² + 3
- Q2: y = 2x² + C → C = 1 → Answer: y = 2x² + 1
- Q3: dy/y = 3 dx → ln y = 3x + C → y = 3e^(3x) → Answer: y = 3e^(3x)
- Q4: dy/y = 1 dx → ln y = 1x + C → y = 3e^(1x) → Answer: y = 3e^(1x)
- Q5: y dy = x dx → y²/2 = x²/2 + C → y² = x² + 9 → Answer: y = √(x² + 9)
- Q6: y dy = x dx → y²/2 = x²/2 + C → y² = x² + 1 → Answer: y = √(x² + 1)
Syllabus note
This skill is listed under "Differential Equations" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is variable separable differential equations in?
"Differential Equations", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.