Solve the separable differential equation dy/dx=3x^2/2y with the initial condition y(0)=4. State the interval on which the solution is valid.

Solve the separable differential equation $\frac{d y}{d x} = \frac{3 x ^{2}}{2 y}$ with the initial condition $y (0) = 4$ . State the interval on which the solution is valid.

easy 2 marks Class 12 · Mathematics · Differential Equations

Answer

The solution is

y^{2} = x^{3} + 16

, valid for all real

x

The correct answer is The solution is $y^{2} = x^{3} + 16$ , valid for all real $x$ . — State the interval: the expression under the root is always positive, so the solution is valid for all $x \in R$ .

Step-by-step solution

Write the equation in separable form: $2 y d y = 3 x^{2} d x$ .
Integrate both sides: $\int 2 y d y = \int 3 x^{2} d x$ .
Compute the integrals: $y^{2} = x^{3} + C$ .
Apply the initial condition $y (0) = 4$ to find $C$ : $4^{2} = 0 + C$ gives $C = 16$ .
Substitute $C$ back: $y^{2} = x^{3} + 16$ .
Take square root (positive branch because $y (0) = 4 > 0$ ): $y = x^{3} + 16$ .
State the interval: the expression under the root is always positive, so the solution is valid for all $x \in R$ .

Common mistake: Forgetting to take the constant of integration after integrating both sides.

Concept tested: Separable first‑order ODE

Published 19 September 2026 · How this solution was prepared and checked

← All Differential Equations questions

Solve the separable differential equation dxdy​=2y3x2​ with the initial condition y(0)=4. State the interval on which the solution is valid.

Step-by-step solution

Solve the separable differential equation $\frac{d y}{d x} = \frac{3 x ^{2}}{2 y}$ with the initial condition $y (0) = 4$ . State the interval on which the solution is valid.