Subject Experts Solutions for Chapter: Differential Equations, Exercise 1: Practice Exercise

What is the solution of the differential equation $\frac{dx}{dy}+\frac{x}{y}-y^2=0$, where $C$ is an arbitrary constant $?$

Consider the following statements

$\mathrm{I}$. The general solution of $\frac{dy}{dx}=f\left(x\right)+x$ is of the form $y=g\left(x\right)+C,$ where $C$ is an arbitrary constant.

$\mathrm{II}$. The degree of ${\left(\frac{dy}{dx}\right)}^2=f\left(x\right)$ is $2$

Which of the above statement $\left(\mathrm{s}\right)$ is$/$are correct $?$

The degree of the differential equation $\frac{dy}{dx}-x={\left(y-x\frac{dy}{dx}\right)}^{-4}$ is

The order and degree of the differential equation of parabolas having vertex at the origin and focus at $(a, 0),$ where $a>0,$ are respectively

Find the solution of $\frac{dy}{dx}=\sqrt{1-x^2-y^2+x^2{y}^2}$, where $C$ is an arbitrary constant.

The differential equation of the family of circles passing through the origin and having centres on the $x$ -axis is

Let $f\left(x\right)$ and $g\left(x\right)$ be twice differentiable functions on $\left[0, 2\right]$ satisfying $f^{\text{'}\text{'}}\left(x\right)=g^{\text{'}\text{'}}\left(x\right), f^{\text{'}}\left(1\right)=4$, $g^{\text{'}}\left(1\right)=6, f\left(2\right)=3$ and $g\left(2\right)=9.$ Then, what is $f\left(x\right)-g\left(x\right)$ at $x=4$ equal to$?$

What are the order and degree respectively of the differential equation whose solution is $y=cx+c^2-3c^{3/2}+2,$ where $c$ is a parameter$?$