Seperable Differential Equations Flashcards
What is a separable differential equation?
A separable differential equation is one that can be expressed in the form dy/dx = g(y)h(x), allowing the variables to be separated.
True or False: Separable differential equations can always be solved by integration.
True
Fill in the blank: A separable differential equation can be rewritten as ____ = g(y)h(x).
dy/dx
What is the first step in solving a separable differential equation?
Separate the variables by rearranging the equation to isolate dy and dx.
Which of the following equations is separable? A) dy/dx = x^2 + y^2 B) dy/dx = xy C) dy/dx = sin(x) + cos(y)
B) dy/dx = xy
What is the general solution of the separable equation dy/dx = 3y?
y = Ce^(3x), where C is a constant.
True or False: The integral of a separable equation can be solved by using partial fractions.
True
When solving dy/dx = y/x, what is the solution after separating variables and integrating?
y = Cx, where C is a constant.
What is the purpose of using initial conditions in separable differential equations?
To determine the specific value of the constant C in the general solution.
Fill in the blank: The solution of dy/dx = ky, where k is a constant, is ____.
y = Ce^(kt)
What technique can be used if the right-hand side of a separable equation is not easily integrable?
You may use substitution or numerical methods.
True or False: The solution to a separable differential equation is unique.
False, it may not be unique depending on initial conditions.
What is the general form of a separable differential equation?
dy/dx = f(y)g(x)
How do you verify if a differential equation is separable?
Check if it can be written as a product of a function of y and a function of x.
What is the integral of dy/(1 + y^2)?
arctan(y) + C
Fill in the blank: For dy/dx = (2x)/(y^2), after separation, we get ____.
y^2 dy = 2x dx
What is the solution to the separable equation dy/dx = y^2?
y = 1/(C - x), where C is a constant.
True or False: The method of separation of variables can be applied to all types of differential equations.
False
What is the role of constants in the solutions of separable differential equations?
Constants represent arbitrary values that can be determined by initial conditions.
What is the solution to the separable equation dy/dx = 3x^2?
y = x^3 + C, where C is a constant.
Fill in the blank: The differential equation dy/dx = (x^3)(y) can be solved by ____.
separation of variables
What does the constant C represent in the solution of a separable differential equation?
It represents the initial condition or arbitrary constant of integration.
If dy/dx = e^x * sin(y), what is the first step to solve it?
Separate the variables to get dy/sin(y) = e^x dx.
What is the solution of the separable equation dy/dx = 1/y?
y = Cx, where C is a constant.
True or False: An equation that cannot be separated is always non-linear.
False
