Homogeneous Differential Equations
Differential Equations
12
JEE Qs
8%
Hard
75
min
Master the systematic approach of recognition, substitution, separation of variables, integration, and back-substitution to reliably solve homogeneous differential equations.
🧮 Key Formulas
✅ Key Points for JEE
- 1Always first check if the given differential equation can be written in the form dy/dx = f(y/x) or dx/dy = g(x/y). This often means all terms in the numerator and denominator have the same degree.
- 2If the equation is of the form dy/dx = f(y/x), substitute y = vx. Differentiate this with respect to x using the product rule to get dy/dx = v + x * dv/dx.
- 3If the equation is of the form dx/dy = g(x/y), substitute x = vy. Differentiate this with respect to y using the product rule to get dx/dy = v + y * dv/dy.
- 4After substitution and simplification, the differential equation will always transform into a variable separable form, which can then be solved by direct integration.
- 5After integrating, remember to substitute back v = y/x (or v = x/y) into the general solution to express it in terms of the original variables x and y.
⚠️ Common Mistakes
- ✕Errors in differentiating the substitution (e.g., product rule for v*x or v*y) leading to incorrect transformed equations.
- ✕Algebraic mistakes during the separation of variables step after substitution, preventing correct integration.
- ✕Forgetting to substitute back v = y/x (or v = x/y) into the final solution, leaving the answer in terms of v, x, and y.
- ✕Incorrectly identifying a non-homogeneous differential equation as homogeneous, leading to futile attempts at substitution.
📝 Practice Questions
See allQ12.Let f : R →R be a twice differentiable function such that f(x + y) = f(x)f(y) for all x, y ∈R. If f ′(0) = 4a and f satisfies f ′′(x) −3af ′(x) −f(x) = 0, a > 0, then the area of the region R = {(x, y) ∣0 ≤y ≤f(ax), 0 ≤x ≤2} is: (1) e2 −1 (2) e2 + 1 (3) e4 + 1 (4) e4 −1
Q24.Let y = f(x) be the solution of the differential equation dydx + x2−1xy = √1−x2x6+4x f(0) = 0. If 6 ∫1/2−1/2 f(x)dx = 2π −α then α2 is equal to _______ .
Q2. Let x = x(y) be the solution of the differential equation y2 dx + (x −1y )dy (1) 1 2 + e (2) 3 + e (3) 3 −e (4) 32 + e
Q9. Let f(x) be a real differentiable function such that f(0) = 1 and f(x + y) = f(x)f ′(y) + f ′(x)f(y) for all x, y ∈R. Then ∑100n=1 loge f(n) is equal to : (1) 2525 (2) 5220 (3) 2384 (4) 2406
Q16.If x = f(y) is the solution of the differential equation (1 + y2) + (x −2etan−1 y) dydx is equal to : f(0) = 1, then f ( √31 ) (1) eπ/12 (2) eπ/4 (3) eπ/3 (4) eπ/6
Q6. Let a curve y = f(x) pass through the points (0, 5) and (loge 2, k). If the curve satisfies the differential equation 2(3 + y)e2xdx −(7 + e2x)dy = 0, then k is equal to (1) 4 (2) 32 (3) 8 (4) 16
NCERT Chapters
- Class 12 Mathematics Ch 9: Differential Equations