Homogeneous, Exact and Linear Equations - One Line Questions
1.
To find the constant of integration g(y) in the potential function F(x, y) = e^x sin y + g(y), we differentiate F with respect to y and equate it to N. —
∂F/∂y = e^x cos y + g'(y) = N
2.
For an exact differential equation M(x, y)dx + N(x, y)dy = 0, the solution is given by F(x, y) = C, where ∂F/∂x = M and which of the following? —
∂F/∂y = N
3.
After finding F(x, y) = x^2y + g(y) for the exact equation (2xy)dx + (x^2 + 3y^2)dy = 0, differentiate F with respect to y and equate to N to find g'(y). —
∂F/∂y = x^2 + g'(y) = x^2 + 3y^2 => g'(y) = 3y^2
4.
If a differential equation is written as M(x, y)dx + N(x, y)dy = 0, what condition makes it an exact differential equation? —
∂M/∂y = ∂N/∂x
5.
Which condition must be satisfied for M(x, y)dx + N(x, y)dy = 0 to be an exact differential equation? —
∂M/∂y = ∂N/∂x
6.
For the exact equation (2xy + 1)dx + (x^2 + 4y)dy = 0, we have M = 2xy + 1 and N = x^2 + 4y. Verify exactness. —
∂M/∂y = 2x, ∂N/∂x = 2x (Exact)
7.
For the equation (e^x sin y)dx + (e^x cos y)dy = 0, what is ∂M/∂y and ∂N/∂x? —
∂M/∂y = e^x cos y, ∂N/∂x = e^x cos y
8.
If a differential equation Mdx + Ndy = 0 is not exact, but M and N are homogeneous functions of the same degree, what is the integrating factor? —
1/(Mx + Ny)
9.
If M and N are homogeneous functions of the same degree n, and Mdx + Ndy = 0 is not exact, what is the integrating factor? —
1/(Mx + Ny)
10.
Consider the linear differential equation dy/dx + (2/x)y = x. What is P(x)? —
2/x
11.
What is the degree of homogeneity of the function f(x, y) = x^3 + y^3? —
3
12.
The function M(x, y) = x^2y is homogeneous of what degree? —
3
13.
What is the general form of a first-order homogeneous differential equation? —
dy/dx = f(y/x)
14.
What is the integrating factor for a first-order linear differential equation of the form dy/dx + P(x)y = Q(x)? —
e^(∫P(x)dx)
15.
Since (e^x sin y)dx + (e^x cos y)dy = 0 is exact, find the potential function F(x, y) by integrating M = e^x sin y with respect to x. —
e^x sin y + g(y)
16.
The differential equation dy/dx = f(x, y) is called homogeneous if f(tx, ty) equals what for any scalar t? —
f(x, y)
17.
If Mdx + Ndy = 0 is exact, and we have found F(x, y) = ∫Mdx + g(y), what is the final solution form? —
F(x, y) = C
18.
If dy/dx = f(x,y) is homogeneous, what is the form of f(tx, ty)? —
f(x,y)
19.
When solving an exact equation Mdx + Ndy = 0, if we integrate M with respect to x to get F(x, y) = ∫Mdx + g(y), what is the relation for g'(y)? —
g'(y) = N - ∂/∂y (∫Mdx)
20.
Integrate g'(y) = 3y^2 to find g(y). —
g(y) = y^3
21.
The equation dy/dx = y/x + tan(y/x) is an example of which type of differential equation? —
Homogeneous
22.
If an equation is exact, M(x, y)dx + N(x, y)dy = 0, and we find a potential function F(x, y) such that ∂F/∂x = M, what is the next step to find F? —
Integrate M with respect to x, treating y as a constant
23.
The differential equation dy/dx = f(x,y) is homogeneous if f(kx, ky) = k^n f(x,y) for some n. What is the specific condition for first-order homogeneous equations as commonly defined? —
n = 0, so f(kx, ky) = f(x, y)
24.
Consider the equation (x+y)dx + (x-y)dy = 0. Is it exact? —
Yes, ∂M/∂y = 1, ∂N/∂x = 1
25.
Consider the equation (x^2 - y^2)dx + 2xy dy = 0. Is this equation exact? —
No, because ∂M/∂y = -2y and ∂N/∂x = 2y
26.
For the differential equation (2xy)dx + (x^2 + y^2)dy = 0, check if it is exact. Here M(x, y) = 2xy and N(x, y) = x^2 + y^2. —
Exact, because ∂M/∂y = 2x and ∂N/∂x = 2x
27.
For the equation (x^2 + y^2)dx + (2xy + y^2)dy = 0, check for exactness. —
Not exact: ∂M/∂y = 2y, ∂N/∂x = 2x
28.
What is the integrating factor for the linear equation dy/dx + (2 tan x)y = sin x? —
sec^2(x)
29.
Let dy/dx = f(y/x). After substituting y=vx, which type of differential equation do we obtain in terms of v and x? —
Separable
30.
Which of the following is NOT a method for solving first-order differential equations? —
Laplace Transform
31.
If Mdx + Ndy = 0 is exact, then the solution can be found by integrating M with respect to x and N with respect to y, then combining terms. —
This statement describes finding the potential function, not the final solution directly.
32.
Calculate the integrating factor for dy/dx + (1/x)y = x^2. —
x
33.
What is the integrating factor for dy/dx + y/x = x? —
x
34.
If dy/dx = f(x,y) is homogeneous, and we substitute y=vx, what is the resulting equation in terms of v and x? —
x dv/dx + v = f(1, v)
35.
Consider the exact equation (2xy)dx + (x^2 + 3y^2)dy = 0. Find the potential function F(x, y) by integrating M=2xy with respect to x. —
x^2y + g(y)
36.
The solution to the exact equation (2xy)dx + (x^2 + 3y^2)dy = 0 is: —
x^2y + y^3 = C
37.
When solving a linear differential equation dy/dx + P(x)y = Q(x), after multiplying by the integrating factor, the left side becomes the derivative of what expression? —
y * (integrating factor)
38.
Solve the linear differential equation dy/dx - 2y = 4. —
y = -2 + Ce^(2x)
39.
Solve the linear differential equation dy/dx + y = e^x. —
y = (1/2)e^x + Ce^(-x)
40.
Solve the linear equation dy/dx - y = e^(2x). —
y = (1/3)e^(2x) + Ce^x
41.
Solve the linear differential equation dy/dx - (1/x)y = x. —
y = (x^2)/2 + Cx
42.
Solve the homogeneous equation dy/dx = y/x. (Hint: y=vx) —
y = Cx
43.
What is the general solution of the homogeneous equation dy/dx = y/x? —
y = Cx
44.
Which substitution is typically used to solve a homogeneous differential equation dy/dx = f(y/x)? —
y = vx
45.
What is the correct substitution for solving the homogeneous equation dy/dx = (x^2 + y^2) / (xy)? —
y = vx
46.
Solve the homogeneous equation dy/dx = (x+y)/x. —
y = x ln(Cx)
47.
Solve the linear equation dy/dx + (1/x)y = 3x. —
y = x^2 + C/x
48.
Consider the differential equation (x^2 + y^2)dx + 2xy dy = 0. Is this equation homogeneous? —
Yes, M and N are homogeneous of degree 2
49.
Consider the differential equation (x^2 + y^2)dx - 2xy dy = 0. Is it homogeneous? —
Yes, M and N are homogeneous of degree 2