Methods for Linear Ordinary Differential Equations with Constant Coefficients

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Questions and Answers

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?

<p>3 (B)</p> Signup and view all the answers

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

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

What does ODE stand for?

<p>Ordinary Differential Equation (C)</p> Signup and view all the answers

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

<p>e^(x^2) (B)</p> 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?

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

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

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

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

<p>Using integrating factor and first-order linear differential equation method (A)</p> 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?

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

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

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

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

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

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

<p>$r_1, r_2 = -1 \pm 3i$ (B)</p> 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?

<p>$y(x) = x^{-1}(c_1\cos(3\ln x) + c_2\sin(3\ln x))$ (A)</p> 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?

<p>u=0,v=y (A)</p> Signup and view all the answers

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

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

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

<p>2nd order (C)</p> Signup and view all the answers

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

<p>e^(x) (C)</p> Signup and view all the answers

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

<p>-1 and -3 (B)</p> Signup and view all the answers

What type of equation involves ordinary derivatives of a function?

<p>Ordinary Differential Equation (C)</p> Signup and view all the answers

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

<p>3rd order (D)</p> Signup and view all the answers

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

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

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

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

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

<p>$y(x) = c_1x^r\cos(r \ln x) + c_2x^r\sin(r \ln x)$ (B)</p> 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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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