Methods for Linear Ordinary Differential Equations with Constant Coefficients

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What is the order of a differential equation?

The degree of the highest derivative present in the equation (correct)
The number of variables in the equation
The number of terms in the equation
The sum of all the coefficients in the equation

Which type of equation involves ordinary derivatives of a function?

Partial Differential Equations
Nonlinear Differential Equations
Linear Differential Equations
Ordinary Differential Equations (correct)

What is the standard form of representing a linear differential equation?

$dx + dy + Py = Q$
$dy / dx + Py = Q$ (correct)
$dy = dx + Py - Q$
$dy^2 / dx^2 + P = Qy$

In a third order differential equation, what is the order of the highest derivative?

3 (B) Signup and view all the answers

Which of the following equations represents a second order differential equation?

$d^2y / dx^2 - 5 + 2y = cos(x)$ (A) Signup and view all the answers

What does ODE stand for?

Ordinary Differential Equation (C) Signup and view all the answers

What is the integrating factor of the differential equation dy/dx = (-3x + 1/x)y + e^(-x)?

e^(x^2) (B) Signup and view all the answers

What is the solution to the differential equation (3x + 2)^2 + 3(3x + 2) - 36y = 3x^2 + 4x + 1?

y(x) = c1(x+2)^2 + c2(x+1)^3 (C) Signup and view all the answers

What is the solution of the differential equation x + y = e^x?

y(x) = (c1 + c2)e^(-x) (C) Signup and view all the answers

How would you solve the differential equation dy/dx + (sec x)y = 7?

Using integrating factor and first-order linear differential equation method (A) Signup and view all the answers

What is the solution to the differential equation dy/dx = [1/(1+x^3)] – [3x^2/(1 + x^2)]y?

$y(x) = e^{\frac{c}{1+x^2}}$ (D) Signup and view all the answers

For the differential equation r(r-1) + 3r-15 = 0, what are the roots?

$r_1 = 5$, $r_2 = -3$ (D) Signup and view all the answers

What are the general solutions to Euler's equation x^2y'' + 3xy' - 15y = 0 with real roots?

$y(x) = c_1x^4 + c_2x^{-5}$ (D) Signup and view all the answers

For the differential equation x^2y'' + 3xy' + 4y = 0, what are the roots of the characteristic equation?

$r_1, r_2 = -1 \pm 3i$ (B) Signup and view all the answers

In solving Euler equations with complex roots, what is the general solution for the equation x^2y'' + 3xy' + 4y = 0?

$y(x) = x^{-1}(c_1\cos(3\ln x) + c_2\sin(3\ln x))$ (A) Signup and view all the answers

What are the functions u and v in the replacement of y by u+v for the second order linear differential equations?

u=0,v=y (A) Signup and view all the answers

What is the standard form of a linear ordinary differential equation?

dy/dx + Py = Q (A) Signup and view all the answers

For a second order linear ordinary differential equation, what is the order of the highest derivative?

2nd order (C) Signup and view all the answers

What is the integrating factor of the differential equation dy/dx = (-3x + 1/x)y + e^(-x)?

e^(x) (C) Signup and view all the answers

In the differential equation x^2y'' + 3xy' + 4y = 0, what are the roots of the characteristic equation?

-1 and -3 (B) Signup and view all the answers

What type of equation involves ordinary derivatives of a function?

Ordinary Differential Equation (C) Signup and view all the answers

In a third order differential equation, what is the order of the highest derivative?

3rd order (D) Signup and view all the answers

What is the integrating factor of the differential equation $\frac{dy}{dx}= \left(-3x+\frac{1}{x}\right)y+e^{-x}$?

$e^{\int \left(-3x+\frac{1}{x}\right)dx}$ (D) Signup and view all the answers

What are the roots of the characteristic equation for the differential equation $x^2y'' + 3xy' + 4y = 0$?

$r = -2, -2i$ (C) Signup and view all the answers

In solving Euler equations with complex roots, what is the general solution for the equation $x^2y'' + 3xy' + 4y = 0$?

$y(x) = c_1x^r\cos(r \ln x) + c_2x^r\sin(r \ln x)$ (B) Signup and view all the answers

What are the functions u and v in the replacement of y by u+v for the second order linear differential equations?

$u = y$, $v = y'$ (A) Signup and view all the answers

What is the general solution for the equation $2x^2y'' + 3xy' - 15y = 0$ with real roots?

$y(x) = c_1x^{r_1} + c_2x^{r_2}$ (D) Signup and view all the answers

What are the roots of the characteristic equation for the differential equation $r(r-1) + 3r-15 = 0$?

$r = 4, -3$ (C) Signup and view all the answers

What is the solution of the differential equation $x + y = e^x$?

$y(x) = cx+e^x-1$ (B) Signup and view all the answers

What is the solution to the differential equation $rac{dy}{dx} + ( ext{sec} x)y = 7$?

$y(x) = e^{\int \text{sec} x dx}(c+7)$ (C) Signup and view all the answers

Solve the LDE: $rac{dy}{dx} = \frac{1}{1+x^3} - \frac{3x^2}{1+x^2}y$.

The standard method yields a general solution involving elementary functions. (A) Signup and view all the answers

Solve the following differential equation: $rac{dy}{dx} = \frac{1}{1+x^3} - \frac{3x^2}{1+x^2}y$.

The standard method yields a general solution involving elementary functions. (B) Signup and view all the answers

Methods for Linear Ordinary Differential Equations with Constant Coefficients

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Questions and Answers

What is the order of a differential equation?

Which type of equation involves ordinary derivatives of a function?

What is the standard form of representing a linear differential equation?

In a third order differential equation, what is the order of the highest derivative?

Which of the following equations represents a second order differential equation?

What does ODE stand for?

What is the integrating factor of the differential equation dy/dx = (-3x + 1/x)y + e^(-x)?

What is the solution to the differential equation (3x + 2)^2 + 3(3x + 2) - 36y = 3x^2 + 4x + 1?

What is the solution of the differential equation x + y = e^x?

How would you solve the differential equation dy/dx + (sec x)y = 7?

What is the solution to the differential equation dy/dx = [1/(1+x^3)] – [3x^2/(1 + x^2)]y?

For the differential equation r(r-1) + 3r-15 = 0, what are the roots?

What are the general solutions to Euler's equation x^2y'' + 3xy' - 15y = 0 with real roots?

For the differential equation x^2y'' + 3xy' + 4y = 0, what are the roots of the characteristic equation?

In solving Euler equations with complex roots, what is the general solution for the equation x^2y'' + 3xy' + 4y = 0?

What are the functions u and v in the replacement of y by u+v for the second order linear differential equations?

What is the standard form of a linear ordinary differential equation?

For a second order linear ordinary differential equation, what is the order of the highest derivative?

What is the integrating factor of the differential equation dy/dx = (-3x + 1/x)y + e^(-x)?

In the differential equation x^2y'' + 3xy' + 4y = 0, what are the roots of the characteristic equation?

What type of equation involves ordinary derivatives of a function?

In a third order differential equation, what is the order of the highest derivative?

What is the integrating factor of the differential equation $\frac{dy}{dx}= \left(-3x+\frac{1}{x}\right)y+e^{-x}$?

What are the roots of the characteristic equation for the differential equation $x^2y'' + 3xy' + 4y = 0$?

In solving Euler equations with complex roots, what is the general solution for the equation $x^2y'' + 3xy' + 4y = 0$?

What are the functions u and v in the replacement of y by u+v for the second order linear differential equations?

What is the general solution for the equation $2x^2y'' + 3xy' - 15y = 0$ with real roots?

What are the roots of the characteristic equation for the differential equation $r(r-1) + 3r-15 = 0$?

What is the solution of the differential equation $x + y = e^x$?

What is the solution to the differential equation $rac{dy}{dx} + ( ext{sec} x)y = 7$?

Solve the LDE: $rac{dy}{dx} = \frac{1}{1+x^3} - \frac{3x^2}{1+x^2}y$.

Solve the following differential equation: $rac{dy}{dx} = \frac{1}{1+x^3} - \frac{3x^2}{1+x^2}y$.

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