91Ó°ÊÓ

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

For each of the differential equations in Exercises 1 to 10 , find the general solution: \(\frac{d y}{d x}=\sin ^{-1} x\)

In each of the Exercises 1 to 10 verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation: \(y=\sqrt{a^{2}-x^{2}} x \in(-a, a): \quad x+y \frac{d y}{d x}=0(y \neq 0)\)

For each of the differential equations in Exercises 1 to 10 , find the general solution: \(e^{x} \tan y d x+\left(1-e^{x}\right) \sec ^{2} y d y=0\)

Form the differential equation of the family of circles having centre on \(y\) -axis and radius 3 units.

Determine order and degree (if defined) of differential equations given in Exercises 1 to 10 . \(y^{\prime \prime}+2 y^{\prime}+\sin y=0\)

Which of the following differential equations has \(y=c_{1} e^{x}+c_{2} e^{-x}\) as the general solution? (A) \(\frac{d^{2} y}{d x^{2}}+y=0\) (B) \(\frac{d^{2} y}{d x^{2}}-y=0\) (C) \(\frac{d^{2} y}{d x^{2}}+1=0\) (D) \(\frac{d^{2} y}{d x^{2}}-1=0\)

For each of the differential equations in Exercises from 11 to 15 , find the particular solution satisfying the given condition: \((x+y) d y+(x-y) d x=0 ; y=1\) when \(x=1\)

Determine order and degree (if defined) of differential equations given in Exercises 1 to 10 . \(y^{\prime \prime}+2 y^{\prime}+\sin y=0\)

The number of arbitrary constants in the general solution of a differential equation of fourth order are: (A) 0 (B) 2 (C) 3 (D) 4