Subtopic: Linear and non-linear first order Ordinary Differential Equations (ODE)
Question Statement:
If $(2xy + 3) dx + (x^2 + 4y) dy = 0$ is an exact differential equation, its general solution is given by $x^2 y + 3x + k y^2 = C$. Calculate the numerical value of the constant $k$.
Answer Key:
2
Step-by-Step Solution & Explanation
Here $M = 2xy + 3$ and $N = x^2 + 4y$.
$\frac{\partial M}{\partial y} = 2x = \frac{\partial N}{\partial x}$
Exact solution:
$\int M dx (\text{with } y \text{ constant}) + \int (N \text{ terms without } x) dy = C$
$\int (2xy + 3) dx + \int 4y dy = C$
$x^2 y + 3x + 2y^2 = C$
Comparing with $x^2 y + 3x + k y^2 = C$, we have $k = 2.0$.