Practice Questions

Name the type of differential equation represented by $\frac{dy}{dx} + Py = Q$, where P and Q are constants or functions of x only.

Identify the order and degree of the differential equation $y' + 5y = 0$.

Identify the order and degree of the differential equation $y'' + (y')^2 + 2y = 0$. Explain your reasoning.

Critique the statement: "The order and degree of a differential equation are always equal." Justify your conclusion with a counterexample.

Solve the differential equation $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$.

What is a 'particular solution' of a differential equation?

Analyze the differential equation $(\frac{d^3y}{dx^3})^2 + 5(\frac{dy}{dx})^4 - \log(x) = 0$ to determine its order and degree.

Explain the difference between a general solution and a particular solution of a differential equation.

Define the order of a differential equation.

The differential equation for a family of curves is given by $\frac{dy}{dx} = \frac{y^2-x^2}{2xy}$. A student solves it and presents the solution as $x^2+y^2 = kx$. Evaluate this proposed solution by checking if it satisfies the differential equation. If it is incorrect, derive the correct solution and justify your steps.

Identify if the differential equation $y' = \sin(x+y)$ is homogeneous. Explain your reasoning.

Explain the concept of an 'ordinary differential equation' and a 'partial differential equation'. Provide one example for each to illustrate the difference.

Identify the order and degree of the differential equation $\left(\frac{d^3y}{dx^3}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^4 + y = 0$. Explain your reasoning.

Summarize the method of 'separation of variables' for solving a first-order, first-degree differential equation.

Describe the complete procedure to solve a homogeneous differential equation of the form $\frac{dy}{dx} = g\left(\frac{y}{x}\right)$. List all the necessary substitutions and steps.

Describe the steps to find the Integrating Factor (I.F.) for a linear differential equation of the form $\frac{dy}{dx} + Py = Q$.

Find the equation of a curve that passes through the point $(1, -1)$ and whose differential equation is $xy \frac{dy}{dx} = (x+2)(y+2)$.

Find the particular solution of the differential equation $\frac{dy}{dx} - 3y \cot x = \sin 2x$, given that $y=2$ when $x = \frac{\pi}{2}$.

A student claims that any homogeneous differential equation of the form $\frac{dy}{dx} = F(\frac{y}{x})$ must have a degree of 1. Evaluate this claim and provide a justification for your conclusion.

Formulate a differential equation to model the temperature $T$ of a body cooling in a room with a constant ambient temperature $T_a$, according to Newton's Law of Cooling, which states that the rate of change of temperature is proportional to the difference between the body's temperature and the ambient temperature. If a body cools from $80^\circ$C to $50^\circ$C in 20 minutes in a room at $20^\circ$C, create a formula for the temperature at time $t$.

Propose a substitution to solve the differential equation $\frac{dy}{dx} = (4x+y+1)^2$. Justify your proposal by transforming the equation into a separable form, and then find the general solution.

Design a differential equation whose general solution represents all non-vertical straight lines passing through the point $(1, 2)$. Solve the created equation to verify your answer.

State the condition for a differential equation of the form $\frac{dy}{dx} = F(x, y)$ to be classified as 'homogeneous'.

Determine the order and degree (if defined) of the differential equation $\frac{d^2y}{dx^2} = \sqrt[3]{1 + (\frac{dy}{dx})^5}$.

Find the general solution of the differential equation $(e^x + 1)y dy = (y+1)e^x dx$.

Examine if the function $y = A e^{2x} + B e^{-2x}$ is a solution to the differential equation $\frac{d^2y}{dx^2} - 4y = 0$.

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

Calculate the integrating factor (I.F.) for the linear differential equation $x \frac{dy}{dx} - 3y = x^3$.

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

Propose a suitable substitution to reduce the differential equation $\frac{dy}{dx} = \cos(x+y)$ to a variable separable form and justify your choice.

An incorrect solution is provided for the differential equation $(x^2 - 3y^2)dx + 2xy dy = 0$. The proposed solution is $x^2 - y^2 = Cx^3$. Critique this solution by checking its validity. Then, formulate the correct approach and derive the correct general solution.

Justify why the degree of the differential equation $\frac{d^2y}{dx^2} + \sin\left(\frac{dy}{dx}\right) = 0$ is not defined, whereas the degree of the equation $\left(\frac{dy}{dx}\right)^2 + \sin y = 0$ is defined.

Solve the differential equation $(x+2y^2)\frac{dy}{dx} = y$ and find the particular solution given that $y=1$ when $x=1$.

Formulate a differential equation representing the family of parabolas with their vertex at the origin and axis along the positive y-axis.

Justify why $e^{\int P dx}$ is chosen as the integrating factor for the linear differential equation $\frac{dy}{dx} + Py = Q$. Derive this factor from the requirement that the left side of the equation becomes an exact derivative.

Explain why the degree of the differential equation $\frac{d^2y}{dx^2} + \cos\left(\frac{dy}{dx}\right) = 0$ is not defined.

Formulate the differential equation for the family of parabolas having their vertex at the origin and axis along the positive y-axis.

Summarize the key characteristics of the three main types of first-order, first-degree differential equations: variable separable, homogeneous, and linear. For each type, describe its standard form and the initial step to solve it.

Propose a method to solve the Bernoulli differential equation $\frac{dy}{dx} + \frac{1}{x}y = x y^2$. Justify your proposed substitution by demonstrating that it transforms the equation into a linear differential equation. Then, find the general solution.

Critique the following reasoning: "To solve $(x+y)dx - (x-y)dy = 0$, we can rearrange it to $\frac{dy}{dx} = \frac{x+y}{x-y}$. Since this is a homogeneous equation, we substitute $y=vx$. This leads to $v + x \frac{dv}{dx} = \frac{x+vx}{x-vx} = \frac{1+v}{1-v}$. Therefore, the solution only depends on the ratio $y/x$." Is the conclusion fully justified? Propose a scenario where a particular solution might not be representable in this form.

Create a differential equation for the family of curves where, for any point $(x,y)$ on the curve, the length of the normal segment between the point and the x-axis is constant and equal to $k$. Solve this differential equation to identify the family of curves and justify your method. (Hint: Length of Normal = $y\sqrt{1 + (y')^2}$)

Solve the differential equation $x^2 dy + (xy+y^2)dx = 0$.

The rate of growth of a bacteria culture is proportional to the number present. If the count was 400 initially and 1000 after 1 hour, what will be the bacteria count after 1.5 hours?

Find the general solution for the differential equation: $y dx - (x+2y^2)dy = 0$.

A tank contains 500 liters of pure water. Brine that contains 0.1 kg of salt per liter is pumped into the tank at a rate of 5 L/min. The well-mixed solution is pumped out at a rate of 10 L/min. Formulate a differential equation for the amount of salt $A(t)$ in the tank at any time $t$. Propose a method to solve this equation and determine the amount of salt in the tank at the moment it becomes empty.