Homogeneous First-Order Differential Equations
A first-order differential equation dy/dx = F(x, y) is homogeneous when the right side can be expressed as a function of y/x or x/y. It is solved by substituting y = vx or x = vy to reduce it to a variable-separable equation.
In one line: Homogeneous means ratio-ready; put y = vx and remember the product-rule v.
Practice This ConceptUnderstand it
In equations of the form dy/dx = f(x, y), homogeneity is checked by seeing whether f(tx, ty) = f(x, y), or whether numerator and denominator are homogeneous expressions of the same degree. With y = vx, y changes with x, so dy/dx = v + x(dv/dx). This converts the equation into one involving v and x, which is usually separable.
Example
Formula and rule
Condition: dy/dx = F(y/x) or dy/dx = F(x/y), or M(x, y)dx + N(x, y)dy = 0 with M and N homogeneous of the same degree.
Main substitution: y = vx, so dy/dx = v + x(dv/dx).
Then solve the resulting separable equation and replace v by y/x.
See it worked out
Follow the example line by line, then use the same method on new questions.
Worked example
Example: Solve dy/dx = (x + y)/x.
- 1Put y = vx, so dy/dx = v + x(dv/dx).
- 2Then v + x(dv/dx) = 1 + v.
- 3Thus x(dv/dx) = 1, so dv = dx/x.
- 4Integrate: v = log|x| + C.
- 5Replace v: y/x = log|x| + C, hence y = x(log|x| + C).
Method you can reuse
- 1Check whether the expression depends on y/x or x/y, or whether numerator and denominator have the same degree.
- 2Choose y = vx or x = vy.
- 3Differentiate using the product rule.
- 4Substitute into the differential equation.
- 5Separate v and x, integrate, add C, and return to x and y.
Visual tip
A diagram is not required for standard CBSE algebraic solving. Homogeneity here means degree pattern in x and y, not visual symmetry of a graph.
Check yourself
Answer the question in your head before opening the answer.
Try it yourself
If y = vx, what is dy/dx?
Think first, then show answerHide answer
Common wrong answer
Treating y = vx as if v were constant, which gives dy/dx = v and destroys the equation. Another common error is not replacing v by y/x in the final answer.
Don't mix this up
Exam tip
Write it in the exam
Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.
How CBSE asks it
- 1
Enough for 1 mark
A first-order differential equation dy/dx = F(x, y) is homogeneous when the right side can be expressed as a function of y/x or x/y. It is solved by substituting y = vx or x = vy to reduce it to a variable-separable equation.
- 2
Add for 2 marks
Example: For dy/dx = (x + y)/x, write y/x = v. Then dy/dx = 1 + v and also dy/dx = v + x(dv/dx). Hence v + x(dv/dx) = 1 + v, so x(dv/dx) = 1, giving v = log|x| + C and y/x = log|x| + C.
- 3
Add for 3 marks
In equations of the form dy/dx = f(x, y), homogeneity is checked by seeing whether f(tx, ty) = f(x, y), or whether numerator and denominator are homogeneous expressions of the same degree.
Practice this concept with focused MCQs
Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.
Help improve this page
Found something confusing, incorrect, or missing?