Solution of homogeneous and linear differential equations of first order - One Line Questions
1.
Integrate (2v / (1 - 3v^2)) dv. Using u-substitution u = 1 - 3v^2, du = 6v dv. The integral is: —
-1/3 ln|1 - 3v^2|
2.
Separate variables for x(dv/dx) = (1 - 3v^2) / (2v). This gives: —
(2v / (1 - 3v^2)) dv = dx/x
3.
Separate variables for x(dv/dx) = (-v^2 - 1) / (v + 1). This gives: —
(v + 1) / (v^2 + 1) dv = -dx/x
4.
Substitute back v = y/x into 1 - 3v^2 = A/x^3. This gives: —
1 - 3y^2/x^2 = A/x^3
5.
Exponentiate both sides: |1 - 3v^2| = e^(-3 ln|x| - 3C) = e^(-3C) * e^(ln|x^-3|) = A * |x^-3|. —
1 - 3v^2 = A*x^-3
6.
Substitute back y = 1/v to find the general solution of dy/dx - y = xy^2. —
1/y = -x + 1 + Ce^-x
7.
Bernoulli's equation dy/dx + P(x)y = Q(x)y^n can be reduced to a linear equation by substituting v = y^(1-n). What is the resulting linear equation? —
dv/dx + (1-n)P(x)v = (1-n)Q(x)
8.
After substitution v = 1/y, the Bernoulli equation dy/dx - y = xy^2 becomes: —
dv/dx + v = -x
9.
What is the standard form of Bernoulli's differential equation? —
dy/dx + P(x)y = Q(x)y^n
10.
Consider the homogeneous equation dy/dx = (x^2 - y^2) / (2xy). What is the form after dividing numerator and denominator by x^2? —
dy/dx = (1 - (y/x)^2) / (2(y/x))
11.
What is the general form of a homogeneous differential equation of the first order? —
dy/dx = f(y/x)
12.
What is the general form of a first-order linear differential equation? —
dy/dx + P(x)y = Q(x)
13.
The differential equation dy/dx + P(x)y = Q(x) is linear. If we multiply by an integrating factor I(x), the equation becomes d/dx(I(x)y) = I(x)Q(x). What is I(x)? —
e^(∫P(x)dx)
14.
What is the integrating factor for the linear differential equation dy/dx + P(x)y = Q(x)? —
e^(∫P(x)dx)
15.
Solve the linear equation dv/dx + v = -x. What is the integrating factor? —
e^x
16.
What type of equation is dy/dx + y/x = x^2? —
Linear
17.
What is the form of equation dy/dx = f(x, y) if f(tx, ty) = f(x, y)? —
Homogeneous
18.
Which type of differential equation can be reduced to a linear form by a suitable substitution? —
Bernoulli's differential equation
19.
For a linear differential equation dy/dx + P(x)y = Q(x), if Q(x) = 0, the equation is called: —
Homogeneous linear
20.
Which method is NOT directly applicable for solving dy/dx + P(x)y = Q(x)? —
Substitution y = vx
21.
Equating the integrals, -1/3 ln|1 - 3v^2| = ln|x| + C. Rearranging gives: —
ln|1 - 3v^2| = -3 ln|x| - 3C
22.
Integrate dx/x. The result is: —
ln|x| + C
23.
What is the condition for a differential equation M(x, y)dx + N(x, y)dy = 0 to be homogeneous? —
M and N are homogeneous functions of the same degree.
24.
After substituting y = vx and rearranging, a homogeneous differential equation usually transforms into a differential equation in which variable(s)? —
x and v
25.
In dy/dx + y/x = x^2, what is P(x) and Q(x)? —
P(x) = 1/x, Q(x) = x^2
26.
For the differential equation dy/dx = f(y/x), after substituting y=vx and differentiating y=vx with respect to x, we get dy/dx = v + x(dv/dx). This is a key step for: —
Solving homogeneous differential equations
27.
Consider the differential equation dy/dx = (x^2 + y^2) / (xy). What is the first step to solve this as a homogeneous equation? —
Rewrite as dy/dx = (1 + (y/x)^2) / (y/x)
28.
Which method is NOT directly applicable for solving dy/dx = f(y/x)? —
Integrating factor method
29.
A first-order differential equation is called linear if it is linear in: —
The dependent variable and its derivative
30.
If we substitute y = vx into a homogeneous differential equation, what does dv/dx represent in terms of x and v? —
The derivative of v with respect to x
31.
The differential equation dy/dx = (x^2 + y^2) / (2xy) can be solved by letting y = vx, which transforms it into a separable equation in x and v. —
True
32.
Consider dy/dx = (x^2 + xy + y^2) / x^2. This is a homogeneous equation because f(tx, ty) = t^2(x^2 + xy + y^2) / t^2x^2 = f(x, y). The degree is 0. —
True, it's homogeneous of degree 0
33.
Solve the linear equation dv/dx + v = -x. What is the general solution for v? —
v = -x + 1 + Ce^-x
34.
Solve the Bernoulli equation dy/dx - y = xy^2. What is the substitution to make it linear? —
v = 1/y
35.
What substitution is used to convert Bernoulli's equation into a linear differential equation? —
v = y^(1-n)
36.
Calculate the integrating factor for dy/dx + y/x = x^2. —
x
37.
Solve the linear differential equation dy/dx + (2/x)y = x. What is the integrating factor? —
x^2
38.
The general solution for dy/dx = (x^2 - y^2) / (2xy) is: —
x^3 - 3xy^2 = C
39.
Simplify x(dv/dx) = (1 - v^2) / (2v) - v. This leads to: —
x(dv/dx) = (1 - 3v^2) / (2v)
40.
Let dy/dx = (1 - (y/x)^2) / (2(y/x)). Substitute y=vx. What is the transformed equation in terms of x and v? —
x(dv/dx) = (1 - v^2) / (2v) - v
41.
Simplify x(dv/dx) = (v - 1) / (v + 1) - v. This leads to: —
x(dv/dx) = (v - 1 - v(v + 1)) / (v + 1) = (-v^2 - 1) / (v + 1)
42.
Consider the homogeneous equation dy/dx = (y - x) / (y + x). After dividing by x, dy/dx = ((y/x) - 1) / ((y/x) + 1). Let y = vx. The equation becomes: —
x(dv/dx) = (v - 1) / (v + 1) - v
43.
Multiply the equation dy/dx + y/x = x^2 by the integrating factor x. The left side becomes the derivative of: —
xy
44.
The equation becomes d/dx(xy) = x^3. Integrate both sides. —
xy = x^4/4 + C
45.
What is the solution of the linear differential equation dy/dx + P(x)y = Q(x) after multiplying by the integrating factor? —
y * I.F. = ∫(Q(x) * I.F.)dx + C
46.
The solution of dy/dx + P(x)y = 0 is: —
y = C * e^(-∫P(x)dx)
47.
Solve the homogeneous differential equation dy/dx = y/x. What is the general solution? —
y = cx
48.
Which substitution is typically used to solve a homogeneous differential equation of the form dy/dx = f(y/x)? —
y = vx
49.
Solve the homogeneous differential equation dy/dx = (x+y)/x. What is the general solution? —
y = x ln|x| + cx
50.
The general solution for dy/dx + y/x = x^2 is: —
y = x^3/4 + C/x