Variable Separable Differential Equations
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Variable Separable Differential Equations

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Questions and Answers

What is the general form of a homogeneous differential equation?

  • An equation that can be expressed as a function of y/x (correct)
  • An equation where terms cannot be grouped by dy and dx
  • An equation containing only variables of x and y with constant coefficients
  • An equation that only has one variable present

    • Which method is commonly used to solve first-order linear ordinary differential equations?

  • Substitution method
  • Separation of variables method
  • Integrating factor method (correct)
  • Graphical method

    • In an initial value problem, what is typically given?

  • A specific value of the function at a certain point (correct)
  • The general solution without any constraints
  • Only the differential equation itself
  • An arbitrary constant with no specific value

    • When using the separation of variables method, what is the first step?

    <p>Rearrange the equation to group all y terms on one side</p> Signup and view all the answers

    What is the outcome of solving the equation $2yy' - (y')^2 = 0$?

    <p>It yields a general solution with arbitrary constants</p> Signup and view all the answers

    Which condition defines a differential equation as homogeneous?

    <p>All terms have the same degree.</p> Signup and view all the answers

    What substitution is typically used for simplifying homogeneous differential equations when $M(x, y)$ is simpler?

    <p>$x = vy$</p> Signup and view all the answers

    In the process of separating variables, how is the equation manipulated?

    <p>Terms involving $y$ are moved to one side and $x$ to the other.</p> Signup and view all the answers

    When integrating with respect to $x$ after separating variables, which technique is useful?

    <p>Direct integration with a constant multiplier.</p> Signup and view all the answers

    In the homogenous equation $ye^{2x}dx = (4 + e^{2x})dy$, which method is applied to separate the variables?

    <p>Dividing both sides by $ye^{2x}$.</p> Signup and view all the answers

    Upon solving the initial value problem using $@ x = 2, y = 3$, what is the value of the constant $C$ determined in the solution?

    <p>5</p> Signup and view all the answers

    What is the form of the final solution for the equation $y^2 + 1 = Cx$ after solving the initial value problem?

    <p>$y^2 + 1 = 5x$</p> Signup and view all the answers

    What is the primary purpose of using integrating factors in differential equations?

    <p>To convert a non-exact equation into an exact equation.</p> Signup and view all the answers

    What is the general form of a second-order linear homogeneous differential equation?

    <p>$y'' + P(x)y' + Q(x)y = 0$</p> Signup and view all the answers

    Which of the following methods is NOT typically used for solving homogeneous differential equations?

    <p>Integrating factors</p> Signup and view all the answers

    In the substitution $y = e^{2x}$ for the equation $y''' - 3y' + 2y = 0$, which derived value is incorrectly calculated?

    <p>$y' = 4e^{2x}$</p> Signup and view all the answers

    What must be satisfied for the solution $y = C_1 ext{cos}( heta t) + C_2 ext{sin}( heta t)$ to be valid for the equation $ rac{d^2x}{dt^2} + heta^2 x = 0$?

    <p>$ heta$ must be greater than zero.</p> Signup and view all the answers

    Which of the following represents the characteristic equation derived from $y'' - 3y' + 2y = 0$?

    <p>$r^2 - 3r + 2 = 0$</p> Signup and view all the answers

    When applying the method of elimination of arbitrary constants, what is typically required?

    <p>The number of equations must match the number of arbitrary constants.</p> Signup and view all the answers

    During the verification process of $e^{2x}$ as a solution to a differential equation, what does substituting yield?

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

    What expression is incorrect for the term $y'''$ for the function $y = C_1e^{3x} + C_2e^{-2x}$?

    <p>$-27C_1e^{3x}$</p> Signup and view all the answers

    Study Notes

    Variable Separable Differential Equation

    • A differential equation is said to be variable separable if all terms containing x and y can be grouped with dx and dy respectively.

    Types of Solutions

    • General solution: Contains an arbitrary constant that is not yet defined.
    • Particular solution: Contains no arbitrary constant. The arbitrary constant has been defined.

    How to Solve Variable Separable Differential Equations

    • Group terms with x and dx on one side, and terms with y and dy on the other side.
    • Integrate both sides after grouping.
    • The solution will usually contain an arbitrary constant, which can be determined if initial conditions are provided.

    Examples

    • Example 1: $y=2x$
      • The general solution is $y=xe^c$, where c is an arbitrary constant.
    • Example 2: $2x(y+1)dx-ydy=0$
      • The particular solution, given the initial condition x = 1, y = 0, is $x^2 - y -ln(y+1) + 1 = c$.

    Homogeneous Differential Equations

    • A differential equation is homogeneous if every term has the same degree.
    • It can be written in the form $M(x, y)dx + N(x, y)dy = 0$, where M and N are functions of x and y with the same degree.
    • Homogeneous differential equations can be solved by making a suitable substitution.

    How to Solve Homogeneous Differential Equations

    • If M is simpler, substitute $x = vy$ and $dx = vdy + ydx$.
    • If N is simpler, substitute $y = vx$ and $dy = vdx + xdv$.

    Elimination of Arbitrary Constant

    • The elimination of arbitrary constant allows us to obtain the differential equation from a given equation.
    • We differentiate the given equation based on the number of arbitrary constants.

    Elimination of Arbitrary Constant Using Elimination Method (Determinant)

    • This method involves constructing a determinant with the given equation and its derivatives.
    • The determinant is then set to zero to obtain the differential equation.

    Examples

    • Example 1: $y = C_1 + C_2e^{3x}$
      • The differential equation is $y'' - 3y' = 0$.
    • Example 2: $2xyy' = 1 + y^2$
      • The particular solution, given the initial condition x = 2, y = 3, is $y^2 + 1 = 5x$.
    • Example 3: $(x^3 + 2x)y dx - x(y^4 + 3)y dy = 0$
      • The particular solution, given the initial condition x = 1, y = 1, is $\frac{1}{3}x^3 + 2ln(x) = \frac{1}{4}y^4 + 3ln(y) + \frac{1}{12}$.
    • Example 4: $ye^{2x}dx = (4+e^{2x})dy$
      • The general solution is $4 + e^{2x} = Cy^2$, where C is an arbitrary constant.

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    Differential Equation Notes PDF

    Description

    Explore the concept of variable separable differential equations with this quiz. Learn how to differentiate between general and particular solutions, and solve examples using integration. Test your understanding with practical applications of these principles.

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