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Separable Differential Equations quiz #1

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  • How do you solve a separable differential equation of the form dy/dx = f(x)g(y)?
    To solve a separable differential equation dy/dx = f(x)g(y), first separate the variables by rewriting as dy/g(y) = f(x)dx. Then, integrate both sides: ∫1/g(y) dy = ∫f(x) dx. Finally, solve for y if possible, and use any initial conditions to find the constant of integration.
  • What is the general solution to the exponential growth and decay differential equation dy/dt = ky?
    The general solution is y = Ce^{kt}, where C is the initial value and k is the growth (k > 0) or decay (k < 0) constant.
  • How is Newton's law of cooling modeled as a separable differential equation, and what is its general solution?
    Newton's law of cooling is modeled by dT/dt = k(T - T_s), where T is the object's temperature and T_s is the surrounding temperature. The general solution is T(t) = (T_0 - T_s)e^{kt} + T_s, where T_0 is the initial temperature.
  • What is the defining characteristic of a separable differential equation?
    A separable differential equation can be written so that dy/dx equals a function of x times a function of y, allowing the variables to be separated on different sides of the equation.
  • What is the first step in solving a separable differential equation?
    The first step is to separate the variables by collecting all terms involving y and dy on one side and all terms involving x and dx on the other.
  • After separating variables in a separable differential equation, what is the next step?
    The next step is to integrate both sides of the equation with respect to their respective variables.
  • How do you use an initial condition to find the constant of integration when solving a separable differential equation?
    You substitute the given values for x and y into the integrated equation and solve for the constant of integration.
  • What is the general solution to the exponential growth and decay differential equation dy/dt = ky?
    The general solution is y = Ce^{kt}, where C is the initial value and k is the growth (k > 0) or decay (k < 0) constant.
  • How is Newton's law of cooling modeled as a separable differential equation, and what is its general solution?
    Newton's law of cooling is modeled by dT/dt = k(T - T_s), and its general solution is T(t) = (T_0 - T_s)e^{kt} + T_s, where T_0 is the initial temperature and T_s is the surrounding temperature.
  • When solving a separable differential equation, what should you do if you cannot explicitly solve for y after integrating?
    If you cannot explicitly solve for y, you leave the solution in its implicit form, as sometimes isolating y is not possible.