Solve the separable differential equation dy/dx=x^3/(1+y^2) and state the domain of the solution.

The implicit solution is y-arctan y = x^4/4+C; the domain is any interval where y is continuous, e.g., (-∞,∞).

Solve the separable differential equation $\frac{d y}{d x} = \frac{x ^{3}}{1 + y ^{2}}$ and state the domain of the solution.

easy 2 marks Class 12 · Mathematics · Differential Equations

Answer

The implicit solution is

y - arctan y = \frac{x ^{4}}{4} + C

; the domain is any interval where

y

is continuous, e.g.,

(- \infty, \infty)

The correct answer is The implicit solution is $y - arctan y = \frac{x ^{4}}{4} + C$ ; the domain is any interval where $y$ is continuous, e.g., $(- \infty, \infty)$ . — State the domain: since the integral of $1/ (1 + y^{2})$ is defined for all real $y$ , the solution is valid for any real $x$ ; thus the domain can be taken as $(- \infty, \infty)$ .

Step-by-step solution

Write the equation as $(1 + y^{2}) d y = x^{3} d x$ by cross‑multiplying.
Integrate both sides: $\int (1 + y^{2}) d y = \int x^{3} d x$ .
Compute the integrals: $y + \frac{y ^{3}}{3} = \frac{x ^{4}}{4} + C$ .
Rewrite the left side as $y - arctan y = \frac{x ^{4}}{4} + C$ using the identity $\frac{d}{d y} (arctan y) = \frac{1}{1 + y ^{2}}$ and rearranging.
State the domain: since the integral of $1/ (1 + y^{2})$ is defined for all real $y$ , the solution is valid for any real $x$ ; thus the domain can be taken as $(- \infty, \infty)$ .

Common mistake: Integrating

1/ (1 + y^{2})

y

instead of

arctan y

Concept tested: Separable differential equation

Published 8 September 2026 · How this solution was prepared and checked

← All Differential Equations questions

Solve the separable differential equation dxdy​=1+y2x3​ and state the domain of the solution.

Step-by-step solution

Solve the separable differential equation $\frac{d y}{d x} = \frac{x ^{3}}{1 + y ^{2}}$ and state the domain of the solution.