Separable Differential Equations quiz #1 Flashcards
Separable Differential Equations quiz #1
You can tap to flip the card.
Control buttons has been changed to "navigation" mode.
1/10Skip to main content
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.