Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
The solutions of $y = x\left(\frac{dy}{dx} + \left(\frac{dy}{dx}\right)^3\right)$ are given by (where $p = \frac{dy}{dx}$ and $k$ is constant)
The constant function $y = 0$
$y = kp^{-3}e^{1/2p^2}(p + p^3)$
$y = kp^3e^{-1/2p^2}(p + p^3)$
$ye^{-1/2p^2} = p^{-2} + 1$
Step-by-Step Solution
Key Concept: Use the substitution $p = \frac{dy}{dx}$ and separate variables after differentiation to reduce the order.
The constant function $y = 0$ is clearly a solution. Differentiating the given equation $\frac{dy}{dx} = (p + p^3) + x(\frac{dp}{dx} + 3p^2\frac{dp}{dx})$ and setting $p = \frac{dy}{dx}$, we get $p^3 - x(1 + 3p^2)\frac{dp}{dx} = -\frac{dx}{x}(\frac{1}{p^3} + \frac{1}{p})dp$. This integrates to $\log xp^3 = \frac{1}{2p^2} + c$, yielding the general solution $xp^3 = ke^{1/2p^2}$.
Correct Answer: 1,2