Ordinary Differential Equations - One Line Questions
1.
What is the particular integral of d²y/dx² - y = x? —
-x
2.
What is the particular integral of d²y/dx² + y = sin(x)? —
-x cos(x)/2
3.
The differential equation M(x,y)dx + N(x,y)dy = 0 is exact if: —
∂M/∂y = ∂N/∂x
4.
What is the integrating factor for the linear differential equation dy/dx + P(x)y = Q(x)? —
e^(∫P(x)dx)
5.
What is the order of the differential equation y'' + 2y' + y = 0? —
2
6.
What is the order of the differential equation dy/dx + y = x? —
1
7.
What is the degree of the differential equation dy/dx + y = x? —
1
8.
What is the degree of the differential equation (dy/dx)³ + x(dy/dx)² + y = 0? —
3
9.
What is the degree of the differential equation d²y/dx² + (dy/dx)² + y = 0? —
1
10.
What is the order of the differential equation (d³y/dx³) + x(d²y/dx²) + (dy/dx) + y = e^x? —
3
11.
What is the degree of the differential equation (d²y/dx²) + x(dy/dx) + y = 0? —
1
12.
What is the order of the differential equation x²(d³y/dx³) + 2x(dy/dx) - y = 0? —
3
13.
The differential equation y'' + y = 0 is an example of: —
A homogeneous linear ODE with constant coefficients
14.
The general solution of y'' + ω²y = 0 is: —
C₁cos(ωx) + C₂sin(ωx)
15.
The differential equation y'' - 4y' + 4y = 0 has a general solution of the form: —
(C₁ + C₂x)e^(2x)
16.
What is the complementary function for the differential equation d²y/dx² - 5dy/dx + 6y = 0? —
C₁e^(2x) + C₂e^(3x)
17.
If the roots of the auxiliary equation of a second-order linear homogeneous ODE are real and distinct, say m₁ and m₂, then the general solution is: —
C₁e^(m₁x) + C₂e^(m₂x)
18.
The general solution of y'' - 2y' + y = 0 is: —
(C₁ + C₂x)e^x
19.
Which of the following is a first-order ordinary differential equation? —
dy/dx + y² = x
20.
The differential equation Mdx + Ndy = 0 is exact if there exists a function F(x,y) such that: —
∂F/∂x = M and ∂F/∂y = N
21.
Which of the following is a Bernoulli differential equation? —
dy/dx + y/x = x²y³
22.
Which differential equation is NOT homogeneous? —
dy/dx = x + y + 1
23.
Which of the following is a homogeneous differential equation? —
dy/dx = x/y + y/x
24.
What is the general solution of dy/dx = e^(x+y)? —
e^(-y) = e^x + C
25.
What is the particular integral for the differential equation d²y/dx² - y = e^(2x)? —
e^(2x)/3
26.
For the differential equation dy/dx + 2xy = x, what is the integrating factor? —
e^(x²)
27.
If the roots of the auxiliary equation are complex conjugates, α ± iβ, then the general solution is: —
e^(αx)(C₁cos(βx) + C₂sin(βx))
28.
What is the integrating factor for dy/dx - y = e^x? —
e^(-x)
29.
The differential equation dy/dx = y/x is: —
Separable
30.
The differential equation (2x + 3y)dx + (3x - y)dy = 0 is: —
Exact
31.
The differential equation dy/dx = f(x) is always: —
Separable
32.
A differential equation is called 'autonomous' if: —
It does not explicitly depend on the independent variable.
33.
The differential equation dy/dx = x²y is: —
Non-linear and separable
34.
What is the condition for the differential equation Mdx + Ndy = 0 to be homogeneous? —
M and N are homogeneous functions of the same degree.
35.
A differential equation is called separable if it can be written in the form: —
f(x)dx = g(y)dy
36.
The differential equation dy/dx + P(x)y = 0 is: —
Homogeneous linear
37.
The differential equation x(dy/dx) + y = x² is: —
Linear and first-order
38.
Which type of differential equation is dy/dx = (x+y)/(x-y)? —
Homogeneous
39.
The differential equation dy/dx = (x+y)² is separable after the substitution: —
u = x+y
40.
To solve a Bernoulli equation dy/dx + P(x)y = Q(x)yⁿ, the standard substitution is: —
u = y^(1-n)
41.
What is the integrating factor for the differential equation dy/dx + (1/x)y = x? —
x
42.
What is the integrating factor for x dy/dx - y = x²? —
1/x
43.
The solution of the differential equation dy/dx = sin(x) with y(0) = 1 is: —
y = -cos(x) + 2
44.
What is the general solution of dy/dx = y? —
y = Ce^x
45.
What is the general solution of the differential equation dy/dx = 0? —
y = c
46.
The general solution of the differential equation dy/dx = y/x is: —
y = Cx
47.
The solution to the initial value problem y' = -xy, y(0) = 1 is: —
y = e^(-x²/2)
48.
The differential equation dy/dx = (x+y+1)/(x+y+2) can be reduced to a separable equation by the substitution: —
u = x+y
49.
What is the general solution of the differential equation dy/dx = k (where k is a constant)? —
y = kx + c
50.
Which of the following is a linear ordinary differential equation? —