91Ó°ÊÓ

In each of the Exercises 1 to 10 , show that the given differential equation is homogeneous and solve each of them. \(\left(x^{2}+x y\right) d y=\left(x^{2}+y^{2}\right) d x\)

In each of the Exercises 1 to 10 verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation: \(y=e^{x}+1\) \(y^{\prime \prime}-y^{\prime}=0\)

Form the differential equation of the family of ellipses having foci on \(y\) -axis and centre at origin.

In each of the Exercises 1 to 10 verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation: \(y-\cos y=x \quad: \quad(y \sin y+\cos y+x) y^{\prime}=y\)

The number of arbitrary constants in the particular solution of a differential equation of third order are: (A) 3 (B) 2 (C) 1 (D) 0

At any point \((x, y)\) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point \((-4,-3)\). Find the equation of the curve given that it passes through \((-2,1)\).

In a culture, the bacteria count is \(1,00,000\). The number is increased by \(10 \%\) in 2 hours. In how many hours will the count reach \(2,00,000\), if the rate of growth of bacteria is proportional to the number present?