Variable Separable and Homogeneous Equations
A separable equation can be written g(y) dy = f(x) dx, and a homogeneous equation becomes separable under the substitution y = vx.
What is Variable Separable and Homogeneous Equations?
A separable equation can be written g(y) dy = f(x) dx, and a homogeneous equation becomes separable under the substitution y = vx.
Key formula / rule: Separate first, then integrate both sides and add a single constant.
Key points
- Compute the general solution of a variable separable equation.
- Apply the substitution y = vx to solve a homogeneous equation.
Common exam trap
Dividing by an expression that may vanish, losing singular solutions.
Definitions
- Term
Variable Separable and Homogeneous Equations
- Meaning
A separable equation can be written g(y) dy = f(x) dx, and a homogeneous equation becomes separable under the substitution y = vx.
- Term
Variable Separable and Homogeneous Equations — explanation
- Meaning
Both methods aim at the same target: get all y with dy and all x with dx. Homogeneity is checked by confirming that the right side depends only on y/x.
Learning objectives
Compute the general solution of a variable separable equation.
Apply the substitution y = vx to solve a homogeneous equation.
Formulae
- Key point
Separate first, then integrate both sides and add a single constant.
- Key point
A function is homogeneous of ° zero if f(tx, ty) = f(x, y).
- Key point
The substitution y = vx converts a homogeneous equation to separable form.
Prerequisites
MAT-U2-DE-T1-S1-C1
Common mistakes
Dividing by an expression that may vanish, losing singular solutions.
Adding a constant of integration on both sides separately.
Keywords
Variable
Separable
Homogeneous
Equations
Practice preview
Find the general solution of the differential equation (x^2 + y^2) dx - 2xy dy = 0.…
medium
Solve the differential equation dy/dx = (1+y^2)/(1+x^2).…
easy
The differential equation dy/dx = (x+y)/x is a homogeneous equation. What substitution is typically used to solve it?…
easy