1 mark, −0.333MathematicsDifference and differential equationsFirst-order differential equationsModerateabout 2 min

IIT JAM 2021 Question 8: Difference and differential equations

The differential equation $(3x^2 y + y^3)\,dx + (x^3 + 3xy^2)\,dy = 0$ is

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Answer and worked solution

Answer: (A)

Write $M = 3x^2 y + y^3$ and $N = x^3 + 3xy^2$ . Both are homogeneous of degree 3, so the equation is homogeneous.

Exactness needs $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$ : here both equal $3x^2 + 3y^2$ , so the equation is also exact. The answer is (A).

The trap: (C) exact and not homogenous

Checking exactness alone can lead to (C); the coefficients are also homogeneous of the same degree, so the equation is both, making (A) correct.

Plain-text version

The differential equation (3x² y + y³) dx + (x³ + 3xy²) dy = 0 is (A) homogenous and exact (B) neither separable nor exact (C) exact and not homogenous (D) homogenous and not exact

Also written as: The differential equation (3x^2 y + y^3) dx + (x^3 + 3xy^2) dy = 0 is

Difference and differential equations has come up 20 times in 5 exams since 2014.

What this question tests

This question is on First-order differential equations, part of Difference and differential equations in Mathematics.

Solving equations in a function and its first derivative.

TestStop has 7 past-paper questions on this subtopic; the nearest in year are below.

IIT JAM 2021 · Q54Let y(x) > 0 solve dy/dx + y = y². If y(ln 2) = 1/3, then y(ln 3) equals _ (round off to 2 decimal places).

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