Question

For the differential equation \(\left(1+\frac{d^2y}{dx^2}\right)^{3/2} = y\frac{d^2y}{dx^2}\) the order, degree and linearity respectively are:

• A. 3, 2 and non-linear
• B. 2, 3 and non-linear
• C. 2, 3 and linear
• D. 3, 2 and linear

Hint:

To find the degree, always make sure the equation is a polynomial in its derivatives first. This means eliminating all fractional powers and denominators involving derivatives.

Answer 1

The Correct Option is B

Step 1: Determine the order of the ODE. The order is the highest derivative present in the equation. Here, the highest derivative is \(\frac{d^2y}{dx^2}\). Thus, the order is 2.
Step 2: Determine the degree of the ODE. The degree is the highest power of the highest-order derivative after the equation has been cleared of radicals and fractions in its derivatives. First, square both sides to remove the \(3/2\) power: \[ \left[ \left(1+\frac{d^2y}{dx^2}\right)^{3/2} \right]^2 = \left(y\frac{d^2y}{dx^2}\right)^2 \] \[ \left(1+\frac{d^2y}{dx^2}\right)^{3} = y^2\left(\frac{d^2y}{dx^2}\right)^2 \] Now, expand the left side using the binomial theorem \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\): \[ 1 + 3\left(\frac{d^2y}{dx^2}\right) + 3\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^3 = y^2\left(\frac{d^2y}{dx^2}\right)^2 \] The highest power of the highest derivative (\(\frac{d^2y}{dx^2}\)) is 3. Thus, the degree is 3.
Step 3: Determine the linearity. An equation is linear if the dependent variable \(y\) and all its derivatives appear only to the first power and are not multiplied together. This equation contains terms like \(\left(\frac{d^2y}{dx^2}\right)^3\) and \(y^2\left(\frac{d^2y}{dx^2}\right)^2\). Because of these higher powers and products, the equation is non-linear. The correct description is order 2, degree 3, and non-linear.