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?

<p>C = 0</p> Signup and view all the answers

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

<p>ω = 5 or ω = -5</p> Signup and view all the answers

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

<p>r = -1 or r = -3</p> Signup and view all the answers

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

<p>C = 0</p> 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?

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

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

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

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

<p>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.</p> 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?

<p>The spring is stretched at $t = 0$ and moving away from the equilibrium position.</p> 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?

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

Which of the ODEs listed is linear?

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

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

<p>The minimum amount of food, M, must satisfy $M &gt; 10$ to ensure positive growth.</p> Signup and view all the answers

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

<p>The mass is farthest from equilibrium when its velocity changes direction after reaching maximum displacement.</p> 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?

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

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

<p>The damping coefficient $c$ must be less than $6$ for oscillation.</p> 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?

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

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

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

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

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

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

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

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

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

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

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

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

<p>The time between successive roots is constant, indicating a predictable oscillatory behavior.</p> 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?

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

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

<p>Since the system is overdamped, the solution does not exhibit oscillatory behavior, leading to a single crossing through equilibrium.</p> 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.

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

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

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

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

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

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

<p>The spring is most stretched when $x(t)$ reaches its maximum positive value, which can be found by analyzing the derivative.</p> 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?

<p>No, the mass will not move slower since the system is undamped and maintains a constant speed through each passage.</p> 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?

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

Flashcards

Equilibrium Solution

A solution to a differential equation that is a constant value, meaning its derivative is always zero.

Method of Undetermined Coefficients

A method used to find solutions to a differential equation by assuming the solution takes a specific form (e.g., xn, tn, ert). The goal is to determine the values of the parameters (n, r) that make the assumed form a valid solution.

Separable Differential Equation

A differential equation where the dependent variable and its derivative are separated on opposite sides of the equation, allowing direct integration.

Specific Solution

A solution that satisfies a specific initial condition (e.g., y(0) = 3).

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Initial Condition

A condition given at a particular point in time, often at the beginning (t = 0).

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Differential Equation

A differential equation that describes the rate of change of a quantity based on the quantity itself and possibly other variables. It can be solved to find the quantity as a function of time.

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Explicit Solution

A solution that is unique to a specific differential equation and satisfies a specific initial condition.

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Non-Separable Differential Equation

A differential equation that is expressed in a way that the dependent variable and its derivative exist on both sides of the equation but are not directly separated.

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Linear Differential Equation (First Order)

A first-order differential equation is called linear if it can be written in the form y'(t) + p(t)y = q(t), where p(t) and q(t) are continuous functions of t.

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Constant Solution

A constant solution of a differential equation is a value of the dependent variable that satisfies the equation for all values of the independent variable.

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Logistic Equation

The logistic equation is a population growth model that takes into account carrying capacity, the maximum population size the environment can sustain.

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Phase Diagram

A phase diagram helps visualize the behavior of a solution to a differential equation without solving it explicitly.

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Critically Damped System

A critically damped system is a type of damped oscillator where the system returns to equilibrium as quickly as possible without oscillating.

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Imaginary Part of a Complex Root

The imaginary part of a complex root of a quadratic equation is given by -b/(2a), where a and b are the coefficients of the quadratic equation.

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Complex Conjugate Roots

The roots of a quadratic equation with negative discriminant are complex conjugates. This means that they have the same real part but opposite imaginary parts.

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Distance between Roots

The distance between successive roots of the solution to the differential equation y'' + 2y' + 2y = 0 is constant, indicating that the solution is not periodic but crosses the axis at regular intervals.

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Position of the mass (t > 0)

The mass's displacement from its equilibrium position at any time t greater than 0.

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Times of direction change

The times when the mass's velocity changes from positive to negative or vice versa, indicating a change in direction.

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Time of maximum compression

The time when the mass is furthest away from its equilibrium position during compression.

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Stretched spring

The spring is stretched when the displacement x(t) is positive, meaning the mass is further away from its equilibrium position than its resting state.

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Compressed spring

The spring is compressed when the displacement x(t) is negative, meaning the mass is closer to its equilibrium position than its resting state.

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Oscillation and passing through equilibrium

The mass will oscillate and pass through the equilibrium position infinitely many times if the system is underdamped. This means the damping coefficient (c) must be less than a certain value.

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Influence of initial conditions on equilibrium passage

The initial conditions (x(0) and x'(0)) determine whether the mass will pass through the equilibrium position infinitely many times, even if the system is critically damped or overdamped.

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Mass farthest from equilibrium

The mass is further away from the equilibrium position when it experiences the maximum amplitude of oscillation. This occurs when the spring is stretched or compressed the most.

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General Solution for Overdamped System

The general solution to the ODE for an overdamped mass-spring-dashpot system, where 'r1' and 'r2' are the roots of the characteristic equation.

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Equilibrium Passage for Overdamped System

In an overdamped mass-spring-dashpot system, the mass can pass through the equilibrium position at most once. It then returns to the equilibrium position asymptotically.

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General Solution for Underdamped System

The general solution to the ODE for an underdamped mass-spring-dashpot system, expressed in terms of 'α' and 'β' where 'r1 = α + iβ' and 'r2 = α - iβ' are the roots of the characteristic equation.

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Equilibrium Passage for Underdamped System

In an underdamped mass-spring-dashpot system, the mass will oscillate and pass through the equilibrium position infinitely many times. The amplitude of these oscillations decreases over time.

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Max Compression and Behaviour for y(t) = 6e^(-2t) - 3te^(-2t)

A mass-spring system with position function y(t) = 6e^(-2t) - 3te^(-2t) is initially stretched since y(0) > 0. The mass reaches its maximum compression at the point where y(t) = -3/2, and then oscillates about the equilibrium position with decreasing amplitude, approaching the equilibrium position asymptotically.

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Equilibrium Passage for x(t) = 2e^(-3t) - 5e^(-2t)

In a mass-spring-dashpot system with position function x(t) = 2e^(-3t) - 5e^(-2t), the mass never passes through the equilibrium position because the function is always positive, indicating that the spring is always stretched.

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Behaviour of a Simple Harmonic Oscillator

The mass in a simple harmonic oscillator (spring-mass) system will oscillate with constant amplitude and frequency, but will never come to rest. The amplitude of the oscillations will be determined by the initial displacement, and the frequency will be determined by the spring constant and the mass.

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Behaviour of a Damped Oscillator with m = 1, c = 4, k = 5

The mass in a spring-mass-dashpot system with m = 1, c = 4, and k = 5, with initial conditions x(0) = 2 and x'(0) = -2, will initially be stretched (since x(0) > 0) and will oscillate about the equilibrium position with decreasing amplitude before eventually coming to rest at the equilibrium position.

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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" = ecosty' - √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 = ert is a solution to: y" + 4y' + 3y = 0

Problem S7

  • Find all equilibrium (constant) solutions to: y'(et + cos(3t − 4)) - y²et = -et (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)

Other Problems

  • S1(Page 2): Find constant solutions and explicit solutions for separable ODEs, including initial conditions. Solve IVP.
  • S2(Page 2): Solve IVP for differential equations involving trigonometric functions (sin t^2 and cosine t^2).
  • S3(Page 2): Find constant solutions and explicit solutions of homogenous equations
  • S1(Page 3): Set up and solve an ODE for a boat with a velocity problem and find how far the boat coasts.
  • S1, S2, S3(Page 3): Address imaginary parts of roots, complex conjugates of roots, and solutions to IVPs.
  • S1, S2(Page 3): Solve various IVP problems, (critical, overdamped systems). Find the general solution to the ODE in terms of r1, r2 using c1 and c2 for the unknown constants, also show how the mass can pass through the equilibrium position.
  • S3(Page 4): Involves underdamped systems. Show that the mass passes through the equilibrium position infinitely many times.
  • S4(Page 4): Analyze a mass-spring-system with its position function.
  • S5(Page 4): Examine a mass-spring-dashpot system, addressing conditions for stretches and compressions, passing through equilibrium, when the spring is most stretched/compressed, and whether it passes through equilibrium.
  • S6(Page 4): Work with an ODE with an initial condition for mass-spring system, determining distance from equilibrium, examining if the mass slows down while passing through equilibrium, and the 4th time vs 1st time.
  • S7(Page 4): Problems are for spring mass-dashpot system, finding positions, times when the mass changes direction, and when the mass is most compressed.
  • S8 and S9 (Page 5): Finding spring positions and examining whether springs are ever stretched/compressed and equilibrium positions.
  • S10 (Page 5): Address spring mass-dashpot system, oscillating conditions, equilibrium, and scenarios involving a changed initial velocity.
  • S11(Page 5): Analyze a spring-mass-dashpot system and calculate position, and identify when the mass passes through equilibrium.
  • S1(Page 6): Analyze resonance using masses attached to a spring.
  • S1, S2 (Page 6): Solve IVPs involving differential equations with specific initial conditions.

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