A first-order differential equation is said to be homogeneous if:
A
It can be written in the form dy/dx = f(x) + g(y)
B
It can be written in the form dy/dx = f(y/x)
C
It has constant coefficients
D
It has no solution
Analysis & Theory
A differential equation of the form dy/dx = f(y/x) is called homogeneous.
Which substitution is generally used to solve a homogeneous differential equation?
A
x = vy
B
y = vx
C
y = x + v
D
x = y + v
Analysis & Theory
The substitution y = vx (or v = y/x) is commonly used to reduce the equation.
The general form of a homogeneous differential equation is:
A
dy/dx = f(x, y)
B
dy/dx = f(y/x)
C
dy/dx + P(x)y = Q(x)
D
M(x,y)dx + N(x,y)dy = 0
Analysis & Theory
A homogeneous equation is of the form dy/dx = f(y/x).
If an equation is homogeneous of degree n, it means:
A
Every term has power n
B
Every term has the same total degree n in x and y
C
The equation has order n
D
The equation has degree n
Analysis & Theory
A function is homogeneous of degree n if all terms have the same total degree in x and y.
Which of the following is a homogeneous equation?
A
dy/dx = (x^2 + y^2)/x^2
B
dy/dx = (x + y)/(x - y)
C
dy/dx = x^2 + y
D
dy/dx = e^x + y
Analysis & Theory
dy/dx = (x + y)/(x - y) can be expressed in terms of y/x, hence homogeneous.
In solving homogeneous equations using substitution y = vx, we get:
A
A linear equation in v and x
B
A separable equation in v and x
C
A quadratic equation in v and x
D
No simplification
Analysis & Theory
After substitution y = vx, dy/dx becomes v + x dv/dx, which leads to a separable form.
The degree of a homogeneous differential equation is always:
A
1
B
Equal to order
C
Greater than order
D
Not defined
Analysis & Theory
Homogeneous differential equations are generally first degree in derivatives.
Which of the following equations is NOT homogeneous?
A
dy/dx = (x^2 - y^2)/(xy)
B
dy/dx = (x + y)/(x - y)
C
dy/dx = x^2 + y^2
D
dy/dx = (y/x) + (1/(y/x))
Analysis & Theory
dy/dx = x^2 + y^2 cannot be reduced to a function of y/x.
What is the usual first step to solve a homogeneous equation dy/dx = f(y/x)?
A
Use substitution y = vx
B
Use substitution x = vy
C
Multiply through by dx
D
Differentiate again
Analysis & Theory
We use y = vx so that dy/dx = v + x dv/dx, reducing it to separable form.
The solution of a homogeneous differential equation usually involves:
A
Logarithmic functions
B
Polynomial functions
C
Trigonometric functions only
D
No integration required
Analysis & Theory
After substitution, separable equations often lead to logarithmic solutions.