Vardaan Learning Institute

Find the order and degree of the differential equation $x^2\left(\frac{d^2y}{dx^2}\right)^3 + y\left(\frac{dy}{dx}\right)^4 + y^4 = 0$.

Determine the degree of the differential equation $\frac{d^2y}{dx^2} + \sin\left(\frac{dy}{dx}\right) = 0$.

Determine the order of the differential equation whose general solution is given by $y = A e^x + B e^{-x}$.

Verify whether $y = e^x$ is a solution of the differential equation $y'' - y = 0$.

Form the differential equation for the family of lines passing through the origin, $y = mx$.

State the number of arbitrary constants in the general solution of a 3rd order differential equation.

State the number of arbitrary constants in a particular solution of a 4th order differential equation.

Find the order and degree of the equation $\left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} - \sin^2 y = 0$.

Write the degree of the differential equation $\sqrt{1 + \left(\frac{dy}{dx}\right)^3} = \frac{d^2y}{dx^2}$.

Show that $y = \sin x + \cos x$ is a solution of the differential equation $y'' + y = 0$.

Solve the differential equation $\frac{dy}{dx} = \frac{x}{y}$.

Solve the differential equation $\frac{dy}{dx} = e^{x+y}$.

Find the general solution of $\frac{dy}{dx} = (1+x^2)(1+y^2)$.

Solve the differential equation $x \frac{dy}{dx} = y$.

Solve the differential equation $\frac{dy}{dx} = \sin x$.

To solve $\frac{dy}{dx} = (x+y)^2$, what substitution should you make to reduce it to variable separable form?

Find the particular solution of $\frac{dy}{dx} = -4xy^2$ given that $y(0) = 1$.

Solve $\sec^2 x \tan y \, dx + \sec^2 y \tan x \, dy = 0$.

Find the general solution of $\frac{dy}{dx} = \sqrt{1-y^2}$.

Solve the differential equation $\frac{dy}{dx} = \frac{1+y}{1+x}$.

Is the differential equation $\frac{dy}{dx} = \frac{x^2+y^2}{xy}$ a homogeneous differential equation? (Yes/No)

For a homogeneous differential equation of the form $\frac{dy}{dx} = F\left(\frac{y}{x}\right)$, what is the standard substitution?

Evaluate the term $v + x \frac{dv}{dx}$ for the homogeneous equation $\frac{dy}{dx} = \frac{x+y}{x}$ after substituting $y=vx$.

Solve the homogeneous differential equation $x \frac{dy}{dx} = x + y$.

Check if $f(x,y) = \sin\left(\frac{y}{x}\right)$ is a homogeneous function. (Yes/No)

Solve the differential equation $\frac{dy}{dx} = \frac{y}{x} + \sin\left(\frac{y}{x}\right)$.

Solve the homogeneous differential equation $2xy \frac{dy}{dx} = x^2 + y^2$.

Identify the standard substitution required to solve $\frac{dx}{dy} = G\left(\frac{x}{y}\right)$.

Solve the differential equation $\frac{dy}{dx} = \frac{x-y}{x+y}$.

Is the differential equation $\frac{dy}{dx} = x + y$ a homogeneous differential equation? (Yes/No)

Write the standard form of a First Order Linear Differential Equation in $y$.

Find the Integrating Factor (I.F.) of $\frac{dy}{dx} + 2y = e^x$.

Solve the linear differential equation $\frac{dy}{dx} + 2y = e^x$.

Find the Integrating Factor (I.F.) of $\frac{dy}{dx} + \frac{y}{x} = x^2$.

Solve the linear differential equation $\frac{dy}{dx} + \frac{y}{x} = x^2$.

Find the Integrating Factor (I.F.) of $x \frac{dy}{dx} + 2y = x^3$. (Hint: Divide by $x$ first).

Find the Integrating Factor (I.F.) of $\frac{dy}{dx} + y\cot x = 2\cos x$.

Find the Integrating Factor (I.F.) of $\frac{dx}{dy} + \frac{x}{y} = y^2$.

Find the Integrating Factor (I.F.) of $(1+x^2)\frac{dy}{dx} + 2xy = \sin x$.

Solve the linear differential equation $\frac{dx}{dy} + x = e^{-y}$.

Write the standard form of Bernoulli's Differential Equation.

To linearize the equation $\frac{dy}{dx} + \frac{y}{x} = y^2$, what substitution should be made after dividing by $y^2$?

Write the differential equation representing a population $P$ that grows at a rate proportional to the population present at time $t$.

Solve the differential equation $\frac{dP}{dt} = 0.05 P$ to find $P(t)$.

Write the differential equation for Newton's Law of Cooling, where $T$ is temperature and $T_s$ is surroundings temperature.

To reduce the equation $\sec^2 y \frac{dy}{dx} + 2x \tan y = x^3$ to a linear DE, what substitution is needed?

If the differential equation of a family of curves is $\frac{dy}{dx} = f(x,y)$, what is the differential equation for its orthogonal trajectories?

Find the orthogonal trajectory for the family of curves satisfying $\frac{dy}{dx} = 2x$.

Solve the differential equation where the slope of the curve at any point $(x,y)$ is equal to $\frac{y}{x}$.

What should you divide by to reduce $\frac{dy}{dx} + y = x y^3$ to a linear differential equation?