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Basics of Differential Equations quiz #1

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  • How do you determine the order and linearity of a differential equation?
    The order of a differential equation is determined by the highest derivative present. To determine if it is linear, check that the dependent variable and its derivatives are not multiplied by each other, are only raised to the first power, and are not inside another function. If all these are true, the equation is linear; otherwise, it is nonlinear.
  • What does it mean for a function to be a solution to a differential equation?
    A function is a solution to a differential equation if, when the function and its derivatives are substituted into the equation, the equation is satisfied (i.e., both sides are equal).
  • How do you find the general and particular solutions to a basic first-order differential equation like y' = 4x^3?
    To find the general solution, integrate both sides to get y = x^4 + C, where C is a constant. To find a particular solution, use an initial condition (such as a point the solution passes through) to solve for C and substitute it back into the general solution.
  • How do you determine the order of a differential equation?
    The order is determined by the highest derivative present in the equation.
  • What are the three criteria for a differential equation to be linear?
    The dependent variable and its derivatives must not be multiplied by each other, must only be raised to the first power, and must not be inside another function.
  • How can you verify if a function is a solution to a differential equation?
    Substitute the function and its derivatives into the equation; if both sides are equal, it is a solution.
  • What is the general solution to the differential equation y' = 4x^3?
    The general solution is y = x^4 + C, where C is a constant of integration.
  • How do you find a particular solution to a differential equation given an initial condition?
    Plug the initial condition into the general solution to solve for the constant C, then substitute C back into the general solution.
  • What does it mean if a differential equation is nonlinear?
    It means at least one of the criteria for linearity is violated, such as variables being multiplied together, raised to powers other than one, or appearing inside another function.
  • What is the difference between a general and a particular solution of a differential equation?
    A general solution includes an arbitrary constant (C), while a particular solution has a specific value for C determined by an initial condition.