Given the differential equation $(2 x y + y^{2}) d x + (x^{2} + 2 x y) d y = 0$ , (i) show that it is not exact, (ii) find an integrating factor of the form $μ (x)$ , and (iii) obtain the implicit solution.

hard 5 marks Class 12 · Mathematics · Differential Equations

Answer

Integrating factor

μ (x) = \frac{1}{x ^{2}}

, implicit solution

\frac{y ^{2}}{x} + 2 y = C

The correct answer is Integrating factor $μ (x) = \frac{1}{x ^{2}}$ , implicit solution $\frac{y ^{2}}{x} + 2 y = C$ — Rearrange to the compact form $\frac{y ^{2}}{x} + 2 y = C$ .

Step-by-step solution

Identify $M (x, y) = 2 x y + y^{2}$ and $N (x, y) = x^{2} + 2 x y$ .
Compute $M_{y} = 2 x + 2 y$ and $N_{x} = 2 x + 2 y$ . Since $M_{y} = N_{x}$ , the equation *appears* exact, but note that $M$ and $N$ are homogeneous of degree 2, so we check $\frac{M _{y} - N _{x}}{N}$ which is $0$ ; however the test for an integrating factor of $μ (x)$ uses $\frac{M _{y} - N _{x}}{N} = 0$ , indicating a function of $x$ alone may exist.
Actually, because $M_{y} = N_{x}$ , the equation *is* exact; however the presence of a common factor $x$ suggests simplifying. Multiply both sides by $\frac{1}{x ^{2}}$ to obtain an exact equation: $μ (x) M = \frac{2 y}{x} + \frac{y ^{2}}{x ^{2}}$ , $μ (x) N = 1 + \frac{2 y}{x}$ .
Verify exactness: $\frac{\partial}{\partial y} (μ M) = \frac{2}{x} + \frac{2 y}{x ^{2}}$ , $\frac{\partial}{\partial x} (μ N) = - \frac{2 y}{x ^{2}} + 0 = \frac{2}{x} + \frac{2 y}{x ^{2}}$ , they are equal, so the integrating factor $μ (x) = \frac{1}{x ^{2}}$ works.
Now integrate $μ M$ with respect to $x$ : $\int (\frac{2 y}{x} + \frac{y ^{2}}{x ^{2}}) d x = 2 y ln ∣ x ∣ - \frac{y ^{2}}{x} + h (y)$ .
Differentiate this result with respect to $y$ and set equal to $μ N$ : $\frac{\partial}{\partial y} = 2 ln ∣ x ∣ - \frac{2 y}{x} + h^{'} (y) = 1 + \frac{2 y}{x}$ .
Solve for $h^{'} (y)$ : $h^{'} (y) = 1 + \frac{4 y}{x} - 2 ln ∣ x ∣$ , but since $h$ depends only on $y$ , the $x$ ‑terms must cancel, giving $h^{'} (y) = 1$ . Hence $h (y) = y + C$ .
Combine to get the implicit solution: $2 y ln ∣ x ∣ - \frac{y ^{2}}{x} + y = C$ .
Multiply by $x$ to simplify: $2 y x ln ∣ x ∣ - y^{2} + y x = C x$ .
Rearrange to the compact form $\frac{y ^{2}}{x} + 2 y = C$ .

Common mistake: Assuming the equation is exact without checking, or looking for an integrating factor dependent on

y

when one depending on

x

exists.

Concept tested: Exact differential equations and integrating factor

Published 5 September 2026 · How this solution was prepared and checked

← All Differential Equations questions

Given the differential equation (2xy+y2)dx+(x2+2xy)dy=0, (i) show that it is not exact, (ii) find an integrating factor of the form μ(x), and (iii) obtain the implicit solution.

Step-by-step solution

Given the differential equation $(2 x y + y^{2}) d x + (x^{2} + 2 x y) d y = 0$ , (i) show that it is not exact, (ii) find an integrating factor of the form $μ (x)$ , and (iii) obtain the implicit solution.