A first-order differential equation is one in which?
A
Only the first derivative appears
B
Second derivative appears
C
No derivative appears
D
All higher-order derivatives appear
Analysis & Theory
A first-order differential equation involves only the first derivative of the unknown function.
Which of the following is a first-order differential equation?
A
d²y/dx² + y = 0
B
dy/dx + y = 0
C
d³y/dx³ + y = 0
D
x² + y² = 1
Analysis & Theory
dy/dx + y = 0 involves only the first derivative.
The general form of a first-order linear differential equation is?
A
dy/dx + P(x)y = Q(x)
B
d²y/dx² + Py = 0
C
dy/dx = f(x, y, dy/dx)
D
dy/dx + y² = 0
Analysis & Theory
The standard form is dy/dx + P(x)y = Q(x).
The solution method for first-order linear equations is known as?
A
Integration by parts
B
Separation of variables
C
Integrating factor method
D
Laplace transform
Analysis & Theory
First-order linear DEs are solved using the integrating factor method.
The differential equation dy/dx = ky has solution?
A
y = kx
B
y = Ce^(kx)
C
y = Ckx
D
y = C + kx
Analysis & Theory
The solution of dy/dx = ky is y = Ce^(kx).
The equation (dy/dx) = (x + y) is?
A
Linear first-order
B
Non-linear first-order
C
Second-order
D
Algebraic
Analysis & Theory
dy/dx = x + y cannot be written in standard linear form; it is non-linear but first-order.
In dy/dx + y = e^x, the function P(x) is?
A
y
B
1
C
e^x
D
x
Analysis & Theory
In standard form dy/dx + P(x)y = Q(x), here P(x) = 1.
In dy/dx + y = e^x, the function Q(x) is?
A
y
B
1
C
e^x
D
x
Analysis & Theory
In standard form dy/dx + P(x)y = Q(x), here Q(x) = e^x.
The integrating factor for dy/dx + y = e^x is?
A
x
B
e^x
C
e^(-x)
D
e^(∫1 dx)
Analysis & Theory
Integrating factor = e^(∫P(x) dx). Here P(x) = 1, so IF = e^(∫1 dx) = e^x.
The solution of dy/dx + y = e^x is?
A
y = Ce^(-x) + (1/2)e^x
B
y = Ce^x + (1/2)e^x
C
y = Ce^(-x) + e^x
D
y = Ce^x + e^x
Analysis & Theory
Using IF = e^x, solution is y·e^x = ∫(e^x·e^x dx) = ∫e^(2x) dx = (1/2)e^(2x) + C. So y = Ce^(-x) + e^x.