The standard form of Bernoulli’s differential equation is:
B
dy/dx + P(x)y = Q(x)y^n
D
d²y/dx² + P(x) dy/dx = 0
Analysis & Theory
Bernoulli’s equation is of the form dy/dx + P(x)y = Q(x)y^n.
If n = 0 or n = 1, Bernoulli’s equation reduces to:
B
Linear differential equation
Analysis & Theory
When n = 0 or 1, Bernoulli’s equation becomes a linear differential equation.
The substitution used to solve Bernoulli’s equation is:
Analysis & Theory
We use z = y^(1-n) to reduce Bernoulli’s equation to a linear form.
After substitution z = y^(1-n), the equation becomes:
B
Linear differential equation in z
Analysis & Theory
The substitution transforms the equation into a linear differential equation in z.
The integrating factor method is applied after substitution because:
A
The equation becomes linear
B
The equation becomes quadratic
C
The equation becomes homogeneous
D
The equation becomes separable
Analysis & Theory
The substitution makes the equation linear in z, so the integrating factor method is used.
Bernoulli’s equation dy/dx + y = y² has:
B
Bernoulli form with n = 2
Analysis & Theory
It matches Bernoulli’s form with P(x) = 1, Q(x) = 1, and n = 2.
The equation dy/dx + (2/x)y = x²y³ is:
A
Linear differential equation
B
Bernoulli’s equation with n = 3
Analysis & Theory
It is of the form dy/dx + P(x)y = Q(x)y^n with n = 3.
In solving Bernoulli’s equation, the role of exponent (1-n) in substitution is:
B
To eliminate non-linearity in y
Analysis & Theory
Exponent (1-n) in substitution z = y^(1-n) eliminates non-linearity in y.
If the original equation is dy/dx + P(x)y = Q(x)y^n, after substitution the equation in z is:
A
dz/dx + (1-n)P(x)z = (1-n)Q(x)
Analysis & Theory
After substitution z = y^(1-n), the reduced equation is dz/dx + (1-n)P(x)z = (1-n)Q(x).
Bernoulli’s equation is classified as:
A
Non-linear differential equation
B
Linear differential equation
C
Exact differential equation
Analysis & Theory
Bernoulli’s equation is non-linear in y, but can be transformed into a linear equation by substitution.