Math 251 Differential Equations Exercises

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Questions and Answers

What integer values of n make y(x) = x^n a solution to xy'' - y' - 3/y = 0?

n = 0 or n = 1

Identify integer values of n for which f(t) = t^n solves d²f/dt² + (1/t)df/dt = 0.

n = 0 or n = -1

For the equation y'' = e^{cos(t)}y' - 2ty' + y^2 - t^2, what constant solutions can be found?

C = 0 or C = t^2/e^{cos(t)}

What constant solutions exist for the equation dφ/dq = q^3φ - dφ/dq?

C = 0 Signup and view all the answers

Determine values of ω for which y = cos(ωt) is a solution of y'' + 25y = 0.

ω = 5 or ω = -5 Signup and view all the answers

What values of r allow y = e^(rt) to solve y'' + 4y' + 3y = 0?

r = -1 or r = -3 Signup and view all the answers

In the equation involving y'' and y(t) = C, which constant values C make the equilibrium solution valid?

C = 0 Signup and view all the answers

What is the explicit solution to the separable ODE y'(t) = y^2t^2 with the initial condition y(0) = 3?

y(t) = 3/(1 - 3t^3) Signup and view all the answers

What is the differential equation governing the position of the mass in S10?

The differential equation is $x'' + cx' + 9x = 0$. Signup and view all the answers

What are the constant solutions for the differential equation given by $\frac{dz}{d\theta} - z^2 \theta^3 e^{\theta} = 7z^2$?

The constant solutions occur when $\frac{dz}{d\theta} = 0$, leading to $z^2(7 + \theta^3 e^{\theta}) = 0$, giving $z = 0$ as the only constant solution. Signup and view all the answers

In S8, what can you conclude about the spring's state at $t = 0$ given x(0) = 1 and x′(0) = 2?

The spring is stretched at $t = 0$ and moving away from the equilibrium position. Signup and view all the answers

For S11, how can it be shown that the mass passes through the equilibrium position every $\frac{\pi}{7}$ seconds?

By demonstrating the motion's pseudo-period and confirming the specified time interval. Signup and view all the answers

Which of the ODEs listed is linear?

The second ODE, $y′(t) - 2e^{-t}y = 4t - 1$, is linear. Signup and view all the answers

What is the minimum amount of food required for the chipmunk population to grow, given $P(0) = 10$?

The minimum amount of food, M, must satisfy $M > 10$ to ensure positive growth. Signup and view all the answers

In S9, when does the mass farthest move from equilibrium?

The mass is farthest from equilibrium when its velocity changes direction after reaching maximum displacement. Signup and view all the answers

After 2 seconds, what is the velocity of the boat if it started at 100 m/s and has a resistive force proportional to the square root of the velocity?

The velocity after 2 seconds is 64 m/s. Signup and view all the answers

What is required for the mass to oscillate infinitely many times in S10?

The damping coefficient $c$ must be less than $6$ for oscillation. Signup and view all the answers

In S7, with m=1, c=3 and k=2, how can we determine that the spring is never stretched?

The mass starts at $x(0) = 0$ with a negative velocity, indicating it moves towards and remains below equilibrium. Signup and view all the answers

If $b^2 - 4ac < 0$ in $ax^2 + bx + c$, what is the imaginary part of the roots?

The imaginary part of the roots is given by $\frac{\sqrt{4ac - b^2}}{2a}$. Signup and view all the answers

Why do the roots of the equation $ax^2 + bx + c$ become complex conjugates when $b^2 - 4ac < 0$?

Because a negative discriminant implies the square root of a negative number, leading to complex roots. Signup and view all the answers

Why is the mass always compressed in S9 with initial conditions $x(0) = 1$ and $x′(0) = 2$?

The mass starts in a stretched position and subsequently moves oscillating around its equilibrium. Signup and view all the answers

What role does the coefficient of damping, $c$, play in the oscillatory behavior as depicted in S10?

The coefficient $c$ determines whether the system is underdamped, overdamped, or critically damped. Signup and view all the answers

What are the initial conditions for the IVP $y'' + 2y' + 2y = 0$?

The initial conditions are $y(0) = 0$ and $y'(0) = 1$. Signup and view all the answers

What can be said about the distance (time) between successive roots of the solution to $y'' + 2y' + 2y = 0$?

The time between successive roots is constant, indicating a predictable oscillatory behavior. Signup and view all the answers

What is the general solution of the characteristic equation for an overdamped mass-spring-dashpot system in terms of roots r1 and r2?

The general solution is given by: $x(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t}$. Signup and view all the answers

Explain why the mass can pass through the equilibrium position at most one time in an overdamped system.

Since the system is overdamped, the solution does not exhibit oscillatory behavior, leading to a single crossing through equilibrium. Signup and view all the answers

Provide the general solution to the ODE for an underdamped mass-spring-dashpot system in terms of α and β.

The general solution is: $x(t) = e^{eta t} (c_1 ext{cos}(eta t) + c_2 ext{sin}(eta t))$. Signup and view all the answers

Why must a mass in an underdamped system pass through the equilibrium position infinitely many times?

The oscillatory nature of the solution allows the mass to oscillate back and forth, crossing the equilibrium point repeatedly. Signup and view all the answers

When does the mass with position function y(t) = 6e^{-2t} - 3te^{-2t} pass through the equilibrium position?

The mass passes through equilibrium when $y(t) = 0$, which can be solved to find specific times. Signup and view all the answers

When is the spring most stretched for the position function x(t) = 2e^{-3t} - 5e^{-2t}?

The spring is most stretched when $x(t)$ reaches its maximum positive value, which can be found by analyzing the derivative. Signup and view all the answers

For the ODE y'' + 9y = 0 with y(0) = -1, will the mass move slower at the 4th passage through equilibrium compared to the 1st time?

No, the mass will not move slower since the system is undamped and maintains a constant speed through each passage. Signup and view all the answers

What can be concluded about the motion of a mass-spring system with initial conditions x(0) = 2 and x'(0) = -2?

The mass will initially be stretched and then start moving towards equilibrium, indicating compression afterwards. Signup and view all the answers

Study Notes

Math 251 Supplemental Exercises

Problem S1-S7: Explore methods for finding specific solutions to differential equations. Methods involve seeking solutions of a particular form.

Problem S1

Find integer values of n for which y(x) = xⁿ is a solution to: xy" - y' = 3y = 0
x*

Problem S2

Find integer values of n for which f(t) = tⁿ is a solution to: d²f/dt² + (1/t) df/dt = 0

Problem S3

Find all equilibrium (constant) solutions to: y" = e^costy' - √2ty' + y² - t² (i.e., find all values of C for which y(t) = C is a solution—where C is a constant)

Problem S4

Find all equilibrium (constant) solutions to: do/dq = q³ dy/dq (i.e., find all values of C for which q(q) = C is a solution—where C is a constant)

Problem S5

Find values of ω for which y = cos(ωt) is a solution to: y" + 25y = 0

Problem S6

Find values of r for which y = e^rt is a solution to: y" + 4y' + 3y = 0

Problem S7

Find all equilibrium (constant) solutions to: y'(e^t + cos(3t − 4)) - y²e^t = -e^t (i.e., find all values of C for which y(t) = C is a solution, where C is a constant) √yy' = 3 + arctan(t² + 1)

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Description

This quiz explores methods for finding specific solutions to differential equations through various exercises. Participants will seek integer values for specific functions and determine equilibrium solutions. Test your knowledge and understanding of differential equations concepts.