Linear Equations and Integration

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Which type of linear equation has a form of $x = a(t)x + f(t)$?

Nonlinear equation

Homogeneous equation

Inhomogeneous equation (correct)

Separable equation

What is the general form of a linear equation with coefficients $b(t)$, $c(t)$, and $g(t)$?

$b(t)x = c(t)x + g(t)$

$x = b(t)x + c(t)x + g(t)$ (correct)

$b(t)x = c(t)x - g(t)$

$x = b(t)x + c(t)x - g(t)$

Which of the following is an example of a nonlinear equation?

$x = t\sin(x)$ (correct)

$y' = 1 - y^2$

$y' = yy'$

$x = (3t + 2)x + t^2 - 1$

What is the general solution to the homogeneous equation $x' = \sin(t)x$?

$x(t) = Ae^{\int \sin(t) dt}$ Signup and view all the answers

What is the constant $A$ in the general solution $x(t) = Ae^{\int \sin(t) dt}$?

A constant that can be positive, negative, or zero Signup and view all the answers

What is the purpose of replacing the constant $e^C$ with the constant $A$ in the general solution $x(t) = Ae^{\int \sin(t) dt}$?

To get rid of the absolute value Signup and view all the answers

What is the important point about linear equations?

The unknown function and its derivative appear alone and only to first order Signup and view all the answers

Which method is used to solve the equation T = −k(T − A)?

Integrating factor Signup and view all the answers

What is the general solution to the linear equation T (t) = A + Ce−kt?

T (t) = A + Ce−kt Signup and view all the answers

What is the integrating factor for the equation x − ax = f?

e− a(t) dt Signup and view all the answers

What is the general solution to the equation x = x + e−t?

x(t) = −1/2 e−t + Cet Signup and view all the answers

What is the integrating factor for the equation x = x sin t + 2te−cos t?

e− sin t dt Signup and view all the answers

What is the particular solution to the equation x = x sin t + 2te−cos t that satisfies x(0) = 1?

x(t) = −1/2 e−t + Ce−t Signup and view all the answers

What are the four steps to solve the equation x = ax + f?

<ol> <li>Rewrite the equation as x − ax = f. 2. Multiply by the integrating factor u(t) = e− a(t) dt. 3. Integrate this equation. 4. Solve for x(t).</li> </ol> Signup and view all the answers

Which equation represents the general solution to the initial value problem in Example 4.25?

$x(t) = (t^2 + e)e^{-\cos t}$ Signup and view all the answers

What is the particular solution to the initial value problem in Example 4.25?

$x(t) = -\cos t^2 + \frac{5},{2}\cos t$ Signup and view all the answers

What is the general solution to the linear equation $y' = -2y + 3$?

$y = \frac{3},{2}e^{2t} + C e^{-2t}$ Signup and view all the answers

What is the particular solution to the linear equation $y' = -2y + 3$ with the initial condition $y(0) = 1$?

$y = \frac{3},{2}e^{-2t} + C e^{2t}$ Signup and view all the answers

What is the general solution to the linear equation $x' = x \tan t + \sin t$?

$x = -\cos t^2 + C \cos t$ Signup and view all the answers

What is the particular solution to the linear equation $x' = x \tan t + \sin t$ with the initial condition $x(0) = 2$?

$x = -\cos t^2 + \frac{5},{2} \cos t$ Signup and view all the answers

What is the method called for solving linear equations by writing an arbitrary solution in the form $y(t) = v(t)y_h(t)$?

Variation of parameters Signup and view all the answers

Which method is used to find an arbitrary solution to the inhomogeneous linear equation $y' = ay + f$?

Method of variation of parameters Signup and view all the answers

What is the form of an arbitrary solution to the inhomogeneous linear equation $y' = ay + f$?

$y(t) = y_h(t) + Ayh(t)$ Signup and view all the answers

What is the difference between a solution to the inhomogeneous equation and a particular solution?

The difference is a constant multiple of the associated homogeneous solution Signup and view all the answers

What is the purpose of the constant $A$ in the general solution $y(t) = yp(t) + Ayh(t)$?

To represent the constant of integration Signup and view all the answers

What is the general form of a linear equation with coefficients $b(t)$, $c(t)$, and $g(t)$?

$y' = b(t)y + c(t)y + g(t)$ Signup and view all the answers

What is the integrating factor for the equation $x' = ax + f$?

$e^{\int a(t) dt}$ Signup and view all the answers

What is the general solution to the linear equation $y' = -2y + 3$?

$y(t) = Ce^{-2t} + 3$ Signup and view all the answers

What is the particular solution to the linear equation $y' = -2y + 3$ with the initial condition $y(0) = 1$?

$y(t) = 2e^{-2t} + 1$ Signup and view all the answers

What is the important point about linear equations?

They have a constant coefficient Signup and view all the answers

Linear Equations and Integration

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Questions and Answers

Which type of linear equation has a form of $x = a(t)x + f(t)$?

What is the general form of a linear equation with coefficients $b(t)$, $c(t)$, and $g(t)$?

Which of the following is an example of a nonlinear equation?

What is the general solution to the homogeneous equation $x' = \sin(t)x$?

What is the constant $A$ in the general solution $x(t) = Ae^{\int \sin(t) dt}$?

What is the purpose of replacing the constant $e^C$ with the constant $A$ in the general solution $x(t) = Ae^{\int \sin(t) dt}$?

What is the important point about linear equations?

Which method is used to solve the equation T = −k(T − A)?

What is the general solution to the linear equation T (t) = A + Ce−kt?

What is the integrating factor for the equation x − ax = f?

What is the general solution to the equation x = x + e−t?

What is the integrating factor for the equation x = x sin t + 2te−cos t?

What is the particular solution to the equation x = x sin t + 2te−cos t that satisfies x(0) = 1?

What are the four steps to solve the equation x = ax + f?

Which equation represents the general solution to the initial value problem in Example 4.25?

What is the particular solution to the initial value problem in Example 4.25?

What is the general solution to the linear equation $y' = -2y + 3$?

What is the particular solution to the linear equation $y' = -2y + 3$ with the initial condition $y(0) = 1$?

What is the general solution to the linear equation $x' = x \tan t + \sin t$?

What is the particular solution to the linear equation $x' = x \tan t + \sin t$ with the initial condition $x(0) = 2$?

What is the method called for solving linear equations by writing an arbitrary solution in the form $y(t) = v(t)y_h(t)$?

Which method is used to find an arbitrary solution to the inhomogeneous linear equation $y' = ay + f$?

What is the form of an arbitrary solution to the inhomogeneous linear equation $y' = ay + f$?

What is the difference between a solution to the inhomogeneous equation and a particular solution?

What is the purpose of the constant $A$ in the general solution $y(t) = yp(t) + Ayh(t)$?

What is the general form of a linear equation with coefficients $b(t)$, $c(t)$, and $g(t)$?

What is the integrating factor for the equation $x' = ax + f$?

What is the general solution to the linear equation $y' = -2y + 3$?

What is the particular solution to the linear equation $y' = -2y + 3$ with the initial condition $y(0) = 1$?

What is the important point about linear equations?

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