Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
For the central conics having their axes along the coordinates:
Differential equation is $y\frac{dy}{dx} = x\left(\frac{dy}{dx}\right)^2 + xy\frac{d^2y}{dx^2}$
Order is 2 and degree is1
Differential equation is $xy\frac{dy}{dx} = x\left(\frac{dy}{dx}\right)^2 + y\left(\frac{d^2y}{dx^2}\right)^3$
Order is 2 and degree is 3
Step-by-Step Solution
Key Concept: Implicit differentiation of the conic equation yields a differential equation relating the function and its derivatives.
For a conic $ax^2 + by^2 = 1$ with axes along coordinate axes, differentiate to obtain $a\frac{dv}{dx} + b\frac{dy}{dx}\frac{d^2y}{dx^2} = 0$. This leads to relationships involving $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ that characterize the conic's differential properties.
Correct Answer: 1,2