2024-241

Consider a particle moving in a one dimensional space. Its acceleration is proportional to the square of its velocity. Also, at \( t=0 \), it has a velocity 2 m/s and at \( t=1 \), it runs at 1 m/s. Consider that the particle starts at \( x=0 \) m.

With that in mind, choose the alternative that correctly describes the particle's position, \( x(t) \), with respect to time.

1. \( x(t) = (\ln(1 + t))/2 \)
2. \( x(t) = 1 + \ln(2 + t^2) \)
3. \( x(t) = 2 + \ln(1 + t^2) \)
4. \( x(t) = 2 \ln( 1 + t ) \)
5. None of the above.

Original idea by: Vitor Antônio Pimenta Silva

2022-168

Choose the correct option regarding the following question related to ordinary differential equations. A racer accelerates from a full stop so that her speed \( t \) seconds after starting is \( 10t \) m/s. How far will her car go in 4 seconds?

1. 40m
2. 60m
3. 80m
4. 180m
5. None of the above

Original idea by: Muhammad Idrees

2022-167

The degree of given differential equation \( y' + y = 5/y' \) will be?

1. 2
2. 3
3. 5
4. 7
5. None of the above

Original idea by: Meer Muhammad Khan

2022-094

A cell has an initial amount of \( l_0 \) liters of water inside it. After one hour has passed, the amount of water is \( 3 l_0 / 4 \). If the rate of water usage for the cell’s physiological functions is inversely proportional to the amount of water in the cell, find an equation that allows you to calculate the amount \( L(t) \) of water in the cell at any point \( t \) in time.

Note: Always consider positive values. Consider time measured in hours, and volume in liters.

1. \( L(t) = l_0 \sqrt{16 - 7t} / 4 \)
2. \( L(t) = l_0 (4 - t) / 4 \)
3. \( L(t) = l_0 \sqrt{-7t^2/16 + 1} / 4 \)
4. \( L(t) = l_0 (t^2 + 16) / 16 \)
5. None of the above

Original idea by: Rómulo Condori

2022-087

Solve the differential equation below for \( y \):

$$ y' - 4 \frac{y}{x} = x $$

Considering \( A \) a constant, choose the alternative that corresponds to a general solution of this equation, if any:

1. \( y = - x^2/3 + A x^4 \)
2. \( y = A x^4 - x^2/2 \)
3. \( y = - x^2/3 - A x^4 \)
4. \( y = A x^4 + x^2/2 \)
5. None of the above

Original idea by: André Portela

2022-085

Consider the following differential equation:

\( x y(x) = (1 + x^2) y'(x) \)

Which alternative represents a family of possible solutions?

1. \( y(x) = c \sqrt{ e^x + 1} \)
2. \( y(x) = c e (x^2 + 1) \)
3. \( y(x) = c \sqrt{ x^2 + 1} \)
4. \( y(x) = c e (x - 1)\)
5. None of the above

Original idea by: Iury Cleveston

2021-043

Choose the alternative that provides a solution for the following differential equation (assume that C is a constant):

$$ y' - 2 x y = x $$

1. \( 2 + 4x^2 \)
2. \( 4x^2 - x^3 \)
3. \( C e^{x^2} - 1/2 \)
4. \( C e^{3x} + 2 \)
5. None of the above

Original idea by: Thales Almeida

2021-042

Consider a function \( f(x) \) given by:

$$ f(x) = - x^\alpha \cos x + C x^\alpha . $$

Choose the value of \( \alpha \) that makes the following formula true.

$$ x f' - 2 f = x^3 \sin x $$

1. \( \alpha = 0 \)
2. \( \alpha = 1 \)
3. \( \alpha = 2 \)
4. \( \alpha = 3 \)
5. None of the above.

Original idea by: Thales Almeida

2021-014

A general solution formula for the equation $$ 2f'(x) = f(x)^2 $$ is:

1. \( f(x) = \frac{2}{x + C} \)
2. \( f(x) = Ce^{x} \)
3. \( f(x) = - \frac{2}{x + C} \)
4. \( f(x) = \ln{x} \)
5. None of the above

Original idea by: Thales Eduardo Nazatto

2021-011

Given the linear differential equation below:

$$ y' = 4y + 16 $$

which alternative corresponds to a possible solution?

1. \( y(t) = e^{4t - 4} - 16 \)
2. \( y(t) = 13e^{4t - 4} - 4 \)
3. \( y(t) = e^{t} - 16 \)
4. \( y(t) = 25e^{t} - 256 \)
5. None of the above.

Original idea by: José Nascimento