A differential equation is said to be of variable separable type if?
A
It can be written as dy/dx + P(x)y = Q(x)
B
It can be written in the form M(x)dx + N(y)dy = 0
C
It involves only second derivatives
D
It cannot be solved by integration
Analysis & Theory
A variable separable DE can be expressed as M(x)dx + N(y)dy = 0.
The standard solution method for variable separable equations is?
A
Laplace transforms
B
Separation of variables and integration
C
Integrating factor
D
Differentiation
Analysis & Theory
Separable equations are solved by separating x and y terms and integrating both sides.
Which of the following is variable separable?
A
dy/dx + y = e^x
B
dy/dx = xy
C
d²y/dx² + y = 0
D
dy/dx + x² = 0
Analysis & Theory
dy/dx = xy can be written as dy/y = x dx, which is separable.
Solve dy/dx = ky (where k is constant). The solution is?
A
y = C + kx
B
y = Ce^(kx)
C
y = kx²
D
y = ln(x) + C
Analysis & Theory
Separating gives dy/y = k dx ⇒ ln y = kx + C ⇒ y = Ce^(kx).
Which of the following is NOT separable?
A
dy/dx = x + y
B
dy/dx = x·y
C
dy/dx = (x²)(y³)
D
dy/dx = (sin x)(cos y)
Analysis & Theory
dy/dx = x + y cannot be written as f(x)g(y).
For dy/dx = (x²)(y²), the separated form is?
A
dy/dx = x² + y²
B
dy/y² = x² dx
C
dy/dx = x²/y²
D
y²dy = dx
Analysis & Theory
dy/dx = x²y² ⇒ dy/y² = x² dx.
The general solution of dy/dx = x/y is?
A
y² = x² + C
B
y = Cx
C
y = Ce^(x²)
D
y² + x² = C
Analysis & Theory
Separating: y dy = x dx ⇒ (1/2)y² = (1/2)x² + C ⇒ y² = x² + C.
dy/dx = (1 + y²)/(1 + x²) is of which type?
A
Separable
B
Linear
C
Non-linear, not separable
D
Exact
Analysis & Theory
dy/dx = (1 + y²)/(1 + x²) ⇒ (dy/(1 + y²)) = (dx/(1 + x²)), which is separable.
In separable equations, after integration we usually get?
A
Only polynomial solutions
B
Implicit or explicit relation between x and y
C
Trigonometric identities
D
No solution
Analysis & Theory
After integration, we get a relation between x and y, which may be implicit or explicit.
The equation dy/dx = (cos x)(cos y) leads to solution?
A
sin y = sin x + C
B
tan y = tan x + C
C
ln|sec y| = ln|sec x| + C
D
sin y = cos x + C
Analysis & Theory
dy/dx = cos x cos y ⇒ dy/cos y = cos x dx ⇒ ∫sec y dy = ∫cos x dx ⇒ ln|sec y + tan y| = sin x + C (equivalent to sin y = sin x + C).