Differential Equations — NDA Mathematics PYQs

What is the solution of the differential equation $\cos \left(\frac{dy}{dx}\right)=p$ when $y(0)=q$ ?

Directions : For the next two (02) items that follow :
Consider the differential equation $e^{x+y}\frac{dy}{dx}=e^{x-y}$ :

What is $\frac{d^{2}y}{d{x}^2}{\left(\frac{dx}{dy}\right)}^2$ equal to ?

Directions : For the next two (02) items that follow :
Consider the differential equation $e^{x+y}\frac{dy}{dx}=e^{x-y}$ :

What is the solution of the differential equation with $y(0)=0$ ?

Consider the following for the two (02) items that follow :
The slope of the tangent to the curve $y=f(x)$ at $(x,f(x))$ is $4$ for every real number $x$ and the curve passes through the origin.

What is the nature of the curve?

What is the degree of the differential equation
$${\left(\frac{d^{2}y}{d{x}^2}\right)}^{\frac{3}{2}}={\left(\frac{dy}{dx}\right)}^{\frac{5}{2}}\text{ ?}$$

Consider the following statements :
I. $y=x{e}^{2x}$ is the solution of $\frac{dy}{dx}=y\left(2+\frac{1}{x}\right)$.
II. $y=x\ln |x|+cx$ is the solution of $\frac{dy}{dx}=\frac{x+y}{x}$.
Which of the statements given above is/are correct?

If $k$ is an arbitrary constant, then what is the general solution of the equation $(x+y{)}^2\frac{dy}{dx}=k^2$?

Let $y_1(x)$ and $y_2(x)$ be two solutions of the differential equation $\frac{dy}{dx}=x$. If $y_1(0)=0$ and $y_2(0)=4$, then what is the number of points of intersection of the curves $y_1(x)$ and $y_2(x)$ ?

The differential equation, representing the curve $y=e^x(a\cos x+b\sin x)$ where $a$ and $b$ are arbitrary constants, is

What are the order and degree respectively of the differential equation
$${\left\{2-{\left(\frac{dy}{dx}\right)}^2\right\}}^{0.6}=\frac{d^{2}y}{d{x}^2}\text{ ?}$$