Question

Identify the statement(s) which is/are true?

• A. The degree and order of differential equation $\sqrt{1+(\frac{d^2y}{dx^2})}=x(\frac{dy}{dx})^{1/3}$ are equal
• B. Solution of the differential equation $xdy-ydx=\sqrt{x^2+y^2}dx$ is $y+\sqrt{x^2+y^2}=Cx^2$.
• C. $\frac{d^2y}{dx^2}=2(\frac{dy}{dx}-y)$ is differential equation of family of curves $y = e^x(A\cos x+B\sin x)$
• D. All the curves possessing the following property, the segment of the tangent between the point of tangency and the x-axis is bisected at the point of intersection with the y-axis are: $x^2 = ky$

Answer 1

Answer: (B), (C)

Statements 2 and 3 are true.

Answer 2

Statement 1 is false because the order is 2 and the degree is 3. Statement 2 is true as derived by solving the homogeneous differential equation. Statement 3 is true as the given differential equation corresponds to the characteristic equation of the given family of curves. Statement 4 is false because the differential equation derived from the property is $\frac{dy}{dx}=\frac{y}{2x}$, which leads to $y^2=kx$, not $x^2=ky$.