Vardaan Learning Institute

What is defined as the order of the highest derivative appearing in a differential equation?

What is the order of the differential equation $\frac{d^2y}{dx^2} + 5\frac{dy}{dx} + 6y = 0$?

What is the order of the differential equation $\left(\frac{dy}{dx}\right)^3 + y = x$?

What is the degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^1 + \left(\frac{dy}{dx}\right)^4 = 0$?

Is the degree defined for the differential equation $\frac{dy}{dx} + \sin\left(\frac{dy}{dx}\right) = 0$? (Yes / No)

If an equation involves an exponential function of a derivative, like $e^{y'} = x$, is its degree defined? (Yes / No)

How many arbitrary constants are there in the general solution of a differential equation of order $n$?

How many arbitrary constants are there in a particular solution of a differential equation of order $n$?

If the general solution of a DE is $y = A \sin x + B \cos x$, what is the order of the differential equation?

State the order and degree of $\sqrt{\frac{dy}{dx}} = x + y$. (Hint: Remove the radical first).

To form a differential equation from a family of curves with 2 arbitrary constants, how many times must you differentiate the equation?

If $y = cx$, differentiate once to find $\frac{dy}{dx}$ in terms of $c$.

In the Variable Separable method, what is the goal before integrating? (i.e., where do $x$ and $y$ terms go?)

Separate the variables for the equation $\frac{dy}{dx} = \frac{x}{y}$.

Integrate both sides of the separated equation: $y \,dy = x \,dx$.

Separate the variables for the differential equation $\frac{dy}{dx} = x^2 y$.

Separate the variables for the equation $\frac{dy}{dx} = e^{x+y}$. (Hint: $e^{x+y} = e^x \cdot e^y$).

Write the integral form of $\frac{dy}{y} = \frac{dx}{x}$.

If a DE is given as $\frac{dy}{dx} = (ax+by+c)^2$, what is the standard substitution used to reduce it to variable separable form?

Separate the variables for $\sec^2 x \tan y \,dx + \sec^2 y \tan x \,dy = 0$.

A function $F(x,y)$ is homogeneous of degree $n$ if $F(\lambda x, \lambda y) = $ _______ $\cdot F(x,y)$.

What is the degree of the homogeneous function $F(x,y) = x^2 + y^2$?

Is the function $F(x,y) = x + \sin y$ a homogeneous function? (Yes / No)

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

If $y = vx$, what is the expression for $\frac{dy}{dx}$ using the product rule?

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

If $x = vy$, what is the expression for $\frac{dx}{dy}$ using the product rule?

Identify if the differential equation $\frac{dy}{dx} = \frac{x+y}{x}$ is homogeneous. (Yes / No)

In the equation from Q28, substitute $y = vx$ into the Right Hand Side $\left(\frac{x+y}{x}\right)$ and simplify.

Is $F(x,y) = e^{y/x}$ a homogeneous function of degree zero? (Yes / No)

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

In the standard form $\frac{dy}{dx} + Py = Q$, what must $P$ and $Q$ be functions of?

What does the acronym I.F. stand for in the context of Linear Differential Equations?

Write the formula for the Integrating Factor (I.F.) for $\frac{dy}{dx} + Py = Q$.

Identify $P$ and $Q$ in the equation: $\frac{dy}{dx} + 2y = \sin x$.

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

Identify $P$ in the equation: $\frac{dy}{dx} + \frac{y}{x} = x^2$.

Find the Integrating Factor (I.F.) for the equation in Q37. (Recall: $e^{\ln x} = x$).

Write the general solution formula for the LDE $\frac{dy}{dx} + Py = Q$ using the I.F.

Write the standard form of a First Order Linear Differential Equation in $x$ (Reverse form).

Write the standard form of Bernoulli's Equation (reducible to linear form).

To solve Bernoulli's Equation $\frac{dy}{dx} + Py = Qy^n$, what do we divide the entire equation by?

After dividing by $y^n$, what substitution is typically made to linearize Bernoulli's equation?

In population growth models, the rate of change of population $P$ with respect to time $t$ is proportional to the current population. Write this as a differential equation.

Separate the variables for the population equation $\frac{dP}{dt} = kP$.

Integrate $\frac{dP}{P} = k \,dt$ to find the general solution for population $P(t)$.

According to Newton's Law of Cooling, the rate of cooling $\frac{dT}{dt}$ is proportional to what temperature difference?

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

If the slope of a curve at any point $(x,y)$ is equal to $2x$, write the corresponding differential equation.

Integrate the equation from Q49 to find the family of curves.