Method of separation of variables - One Line Questions
1.
Integrating dx/x^2 gives: —
-1/x
2.
Consider dy/dx = (1+x^2)/(1-y^2). Separating variables yields: —
(1-y^2) dy = (1+x^2) dx
3.
For dy/dx = x/(1+y), the separated form is: —
(1+y) dy = x dx
4.
For the differential equation dy/dx = (x+1)/(y-1), the separated form is: —
(y-1) dy = (x+1) dx
5.
If dy/dx = (x^2+1)/(y+2), what is the separated form? —
(y+2) dy = (x^2+1) dx
6.
Consider dy/dx = (y^2 - 1)/(x+1). If y=1 or y=-1, these are constant solutions. What is the separated form for y not equal to 1 or -1? —
dy/(y^2 - 1) = dx/(x+1)
7.
Consider dy/dx = cos(y). What is the separated form? —
dy/cos(y) = dx
8.
What is the general form of a first-order differential equation that can be solved using the method of separation of variables? —
dy/dx = f(x)g(y)
9.
Consider the differential equation dy/dx = sec(x)tan(y). The separated form is: —
dy/tan(y) = sec(x) dx
10.
For dy/dx = y/x, the separated form is: —
dy/y = dx/x
11.
Consider dy/dx = y/x^2. The separated form is: —
dy/y = dx/x^2
12.
After separating the variables in dy/dx = f(x)g(y) as dy/g(y) = f(x) dx, what is the next general step? —
Integrate dy/g(y) and f(x) dx separately
13.
When using the method of separation of variables for dy/dx = f(x)g(y), what is the first step in the integration process? —
Integrate dy/g(y)
14.
What is the primary advantage of the method of separation of variables? —
It can solve a wide range of first-order ODEs efficiently.
15.
What is the main limitation of the method of separation of variables? —
It does not work if the variables cannot be separated.
16.
What is the integral of sec(x) dx? —
ln|sec(x) + tan(x)|
17.
What is the integral of dy/cos(y)? —
ln|tan(y/2 + pi/4)|
18.
What is the integral of dy/tan(y)? —
ln|sin(y)|
19.
The general solution to dy/dx = sec(x)tan(y) is: —
ln|sin(y)| = ln|sec(x) + tan(x)| + C
20.
The integral of dx is x. Therefore, the general solution to dy/dx = cos(y) is: —
ln|tan(y/2 + pi/4)| = x + C
21.
What is the integral of dx/(x+1)? —
ln|x+1|
22.
Integrating dx/x gives: —
ln|x|
23.
To integrate dy/(y^2 - 1), we typically use partial fractions. The result involves terms like: —
ln|y-1| and ln|y+1|
24.
Integrating dy/y gives: —
ln|y|
25.
Integrating dy/y gives: —
ln|y|
26.
The general solution to dy/dx = y/x^2 is: —
ln|y| = -1/x + C
27.
The general solution to dy/dx = y/x is: —
ln|y| = ln|x| + C, which simplifies to y = Cx
28.
For the equation dy/dx = xy, the separated form is dy/y = x dx. Integrating both sides gives: —
ln|y| = x^2/2 + C
29.
If dy/dx = cos(x)sin(y), what is the separated form of the equation? —
dy/sin(y) = cos(x) dx
30.
If a differential equation can be written as dy/dx = f(x)g(y), and g(y) is never zero, what happens to the solution if g(y) = 0 for some value of y? —
That value of y is a constant solution (equilibrium solution).
31.
Integrating (1+x^2) dx gives: —
x + x^3/3
32.
Integrating x dx gives: —
x^2/2
33.
Integrating (x+1) dx results in: —
x^2/2 + x
34.
Integrating (x^2+1) dx gives: —
x^3/3 + x
35.
Integrating (1-y^2) dy gives: —
y - y^3/3
36.
The general solution to dy/dx = (1+x^2)/(1-y^2) is obtained by integrating both sides. What is the form of the solution? —
y - y^3/3 = x + x^3/3 + C
37.
Integrating (1+y) dy gives: —
y + y^2/2
38.
The general solution to dy/dx = x/(1+y) is: —
y + y^2/2 = x^2/2 + C
39.
What is the general solution to dy/dx = y^2? —
y = 1/(x + C)
40.
What is the general solution to dy/dx = xy, after exponentiating? —
y = Ce^(x^2/2)
41.
Solve the differential equation dy/dx = y. The solution is of the form: —
y = Ce^x
42.
The differential equation dy/dx = e^x is an example of a separable equation where g(y) = 1. What is its general solution? —
y = e^x + C
43.
What is the general solution to the differential equation dy/dx = 1? —
y = x + C
44.
Find the general solution of dy/dx = x^2. —
y = x^3/3 + C
45.
Consider the differential equation dy/dx = x/y. Which of the following is the correct separation of variables? —
y dy = x dx
46.
Integrating (y-1) dy results in: —
y^2/2 - y
47.
The general solution to dy/dx = (x+1)/(y-1) is: —
y^2/2 - y = x^2/2 + x + C
48.
Integrating (y+2) dy gives: —
y^2/2 + 2y
49.
The general solution to dy/dx = (x^2+1)/(y+2) is: —
y^2/2 + 2y = x^3/3 + x + C