What is the difference between a homogeneous and a non-homogeneous differential equation?

Homogeneous differential equation: Every term is a function of the same degree in ( x ) and ( y ). Non-homogeneous differential equation: Contains terms that are not of the same degree or includes an explicit function of ( x ) (not just in terms of ( y/x )).

What is the solution form of a homogeneous differential equation?

The solution often involves integrating after substitution and may be implicit or explicit in ( x ) and ( y ).

Can higher-order differential equations be homogeneous?

Yes. A higher-order ODE is homogeneous if all terms are of the same degree in the dependent and independent variables and their derivatives.

Are homogeneous differential equations always linear?

No. Homogeneous differential equations can be nonlinear; the term "homogeneous" refers to the degree of the terms, not linearity.

What are some common applications of homogeneous differential equations?

They appear in: Physics (e.g., rate problems, mixing, cooling) Engineering (e.g., circuits, control systems) Population dynamics Economics (growth models)

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Homogeneous Differential Equations

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Homogeneous Differential Equations: Definition, Formula & Examples

1.0What is a Homogeneous Differential Equation?

Homogeneous Differential Equations Definition

A homogeneous differential equation is a first-order differential equation of the form:

$\frac{d y}{d x} = f (x, y)$

where the function f(x,y) is a homogeneous function of degree 0.

In other words, f(x, y) depends only on the ratio y/x (or x/y).

Mathematically, a function f(x, y) is homogeneous of degree n if:

$f (t x, t y) = t^{n} f (x, y), \forall t \neq = 0$

So, if n = 0:

f(tx,ty) = f(x,y)

This ensures the equation can be solved by substitution.

General Form

The general form of a homogeneous differential equation is:

$\frac{d y}{d x} = \frac{F ( x , y )}{G ( x , y )}$

Or in differential form:

$M (x, y) d x + N (x, y) d y = 0$

where both M(x,y) and N(x,y) are homogeneous functions of the same degree.

Condition for Homogeneity

For the equation:

$\frac{d y}{d x} = \frac{F ( x , y )}{G ( x , y )}$

to be homogeneous:

$F (t x, t y) = t^{n} F (x, y), G (t x, t y) = t^{n} G (x, y)$

This ensures both numerator and denominator have the same degree.

2.0Homogeneous Differential Equations Formula

The key formula to solve is:

$y = vx, \frac{d y}{d x} = v + x \frac{d v}{d x}$

$⟹ \frac{d v}{d x} = \frac{f ( v ) - v}{x}$

Thus,

$\int \frac{d v}{f ( v ) - v} = ln ∣ x ∣ + C$

3.0Steps to Solve Homogeneous Differential Equation

The standard substitution method makes solving these equations easier.

Substitution Method (y=vx)

We substitute:

$y = vx ⟹ v = \frac{y}{x}$

Differentiating:

$\frac{d y}{d x} = v + x \frac{d v}{d x}$

This reduces the equation into a separable form.

Derivation of Formula

Start with: $\frac{d y}{d x} = f (\frac{y}{x})$
Substituting y=vx: $\frac{d y}{d x} = v + x \frac{d v}{d x}$
Substitution yields: $v + x \frac{d v}{d x} = f (v)$
Rearranging: $x \frac{d v}{d x} = f (v) - v$
Separable form: $\frac{d v}{f ( v ) - v} = \frac{d x}{x}$
Integrate both sides: $\int \frac{d v}{f ( v ) - v} = ln ∣ x ∣ + C$

General solution obtained.

4.0How to Solve a Homogeneous Differential Equation?

To solve a homogeneous differential equation, follow these systematic steps:

Check for Homogeneity:
Ensure the equation can be written in the form $(\frac{d y}{d x} = F (\frac{y}{x}))$ or that both functions (M(x, y)) and (N(x, y)) are homogeneous of the same degree.
Use the Substitution (y = vx):
Let (y = vx), where (v) is a function of (x). Then, $(\frac{d y}{d x} = v + x \frac{d v}{d x})$ .
Transform the Equation:
Substitute (y) and $(\frac{d y}{d x})$ in the original equation. This should result in an equation involving only (v) and (x).
Separate Variables:
Rearrange the equation to group all (v) terms on one side and all (x) terms on the other, making it separable.
Integrate Both Sides:
Integrate with respect to their respective variables. Use partial fractions or other integration techniques as needed.
Back-substitute:
Replace (v) with (\frac{y}{x}) to return to the original variables.
General Solution:
Simplify the result and include the constant of integration.

5.0Homogeneous Differential Equations Examples (Solved)

Example 1:

Solve

$\frac{d y}{d x} = \frac{x + y}{x - y}$

Solution:

$\frac{d y}{d x} = \frac{x + y}{x - y}$

Put y = vx, dy/dx = v + x dv/dx:

$v + x \frac{d v}{d x} = \frac{1 + v}{1 - v}$

$x \frac{d v}{d x} = \frac{1 + v}{1 - v} - v$

$x \frac{d v}{d x} = \frac{1 + v - v ( 1 - v )}{1 - v} = \frac{1 + v ^{2}}{1 - v}$

So,

$\frac{d v}{d x} = \frac{1 + v ^{2}}{x ( 1 - v )}$

Rearranging:

$\frac{( 1 - v )}{1 + v ^{2}} d v = \frac{d x}{x}$

Integrating:

$\int \frac{1 - v}{1 + v ^{2}} d v = ln ∣ x ∣ + C$

Split:

$\int \frac{1}{1 + v ^{2}} d v - \int \frac{v}{1 + v ^{2}} d v = ln ∣ x ∣ + C$

$tan^{- 1} v - \frac{1}{2} ln (1 + v^{2}) = ln ∣ x ∣ + C$

Back-substitute v=y/x:

$tan^{- 1} (\frac{y}{x}) - \frac{1}{2} ln (x^{2} + y^{2}) = ln ∣ x ∣ + C$

Example 2:

Solve

$(x^{2} + y^{2}) d y = 2 x y d x$

Solutions:

$\frac{d y}{d x} = \frac{2 x y}{x ^{2} + y ^{2}}$

Put y=vx:

$v + x \frac{d v}{d x} = \frac{2 v}{1 + v ^{2}}$

$x \frac{d v}{d x} = \frac{2 v}{1 + v ^{2}} - v$

$x \frac{d v}{d x} = \frac{2 v - v ( 1 + v ^{2} )}{1 + v ^{2}} = \frac{v - v ^{3}}{1 + v ^{2}}$

$\frac{1 + v ^{2}}{v ( 1 - v ^{2} )} d v = \frac{d x}{x}$

6.0Practice Questions

Solve: $\frac{d y}{d x} = \frac{x ^{2} + y ^{2}}{2 x y}$
Solve: $(x^{2} - y^{2}) d y = 2 x y d x$
Show and solve: $\frac{d y}{d x} = \frac{x + y}{x - y}$
Solve: $\frac{d y}{d x} = \frac{3 x ^{2} + y ^{2}}{2 x y}$
Evaluate the solution for: $\frac{d y}{d x} = \frac{x ^{2} - x y}{y ^{2} - x y}$

Frequently Asked Questions

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Homogeneous Differential Equations: Definition, Formula & Examples

1.0What is a Homogeneous Differential Equation?

Homogeneous Differential Equations Definition

A homogeneous differential equation is a first-order differential equation of the form:

$\frac{d y}{d x} = f (x, y)$

where the function f(x,y) is a homogeneous function of degree 0.

In other words, f(x, y) depends only on the ratio y/x (or x/y).

Mathematically, a function f(x, y) is homogeneous of degree n if:

$f (t x, t y) = t^{n} f (x, y), \forall t \neq = 0$

So, if n = 0:

f(tx,ty) = f(x,y)

This ensures the equation can be solved by substitution.

General Form

The general form of a homogeneous differential equation is:

$\frac{d y}{d x} = \frac{F ( x , y )}{G ( x , y )}$

Or in differential form:

$M (x, y) d x + N (x, y) d y = 0$

where both M(x,y) and N(x,y) are homogeneous functions of the same degree.

Condition for Homogeneity

For the equation:

$\frac{d y}{d x} = \frac{F ( x , y )}{G ( x , y )}$

to be homogeneous:

$F (t x, t y) = t^{n} F (x, y), G (t x, t y) = t^{n} G (x, y)$

This ensures both numerator and denominator have the same degree.

2.0Homogeneous Differential Equations Formula

The key formula to solve is:

$y = vx, \frac{d y}{d x} = v + x \frac{d v}{d x}$

$⟹ \frac{d v}{d x} = \frac{f ( v ) - v}{x}$

Thus,

$\int \frac{d v}{f ( v ) - v} = ln ∣ x ∣ + C$

3.0Steps to Solve Homogeneous Differential Equation

The standard substitution method makes solving these equations easier.

Substitution Method (y=vx)

We substitute:

$y = vx ⟹ v = \frac{y}{x}$

Differentiating:

$\frac{d y}{d x} = v + x \frac{d v}{d x}$

This reduces the equation into a separable form.

Derivation of Formula

Start with: $\frac{d y}{d x} = f (\frac{y}{x})$
Substituting y=vx: $\frac{d y}{d x} = v + x \frac{d v}{d x}$
Substitution yields: $v + x \frac{d v}{d x} = f (v)$
Rearranging: $x \frac{d v}{d x} = f (v) - v$
Separable form: $\frac{d v}{f ( v ) - v} = \frac{d x}{x}$
Integrate both sides: $\int \frac{d v}{f ( v ) - v} = ln ∣ x ∣ + C$

General solution obtained.

4.0How to Solve a Homogeneous Differential Equation?

To solve a homogeneous differential equation, follow these systematic steps:

Check for Homogeneity:
Ensure the equation can be written in the form $(\frac{d y}{d x} = F (\frac{y}{x}))$ or that both functions (M(x, y)) and (N(x, y)) are homogeneous of the same degree.
Use the Substitution (y = vx):
Let (y = vx), where (v) is a function of (x). Then, $(\frac{d y}{d x} = v + x \frac{d v}{d x})$ .
Transform the Equation:
Substitute (y) and $(\frac{d y}{d x})$ in the original equation. This should result in an equation involving only (v) and (x).
Separate Variables:
Rearrange the equation to group all (v) terms on one side and all (x) terms on the other, making it separable.
Integrate Both Sides:
Integrate with respect to their respective variables. Use partial fractions or other integration techniques as needed.
Back-substitute:
Replace (v) with (\frac{y}{x}) to return to the original variables.
General Solution:
Simplify the result and include the constant of integration.

5.0Homogeneous Differential Equations Examples (Solved)

Example 1:

Solve

$\frac{d y}{d x} = \frac{x + y}{x - y}$

Solution:

$\frac{d y}{d x} = \frac{x + y}{x - y}$

Put y = vx, dy/dx = v + x dv/dx:

$v + x \frac{d v}{d x} = \frac{1 + v}{1 - v}$

$x \frac{d v}{d x} = \frac{1 + v}{1 - v} - v$

$x \frac{d v}{d x} = \frac{1 + v - v ( 1 - v )}{1 - v} = \frac{1 + v ^{2}}{1 - v}$

So,

$\frac{d v}{d x} = \frac{1 + v ^{2}}{x ( 1 - v )}$

Rearranging:

$\frac{( 1 - v )}{1 + v ^{2}} d v = \frac{d x}{x}$

Integrating:

$\int \frac{1 - v}{1 + v ^{2}} d v = ln ∣ x ∣ + C$

Split:

$\int \frac{1}{1 + v ^{2}} d v - \int \frac{v}{1 + v ^{2}} d v = ln ∣ x ∣ + C$

$tan^{- 1} v - \frac{1}{2} ln (1 + v^{2}) = ln ∣ x ∣ + C$

Back-substitute v=y/x:

$tan^{- 1} (\frac{y}{x}) - \frac{1}{2} ln (x^{2} + y^{2}) = ln ∣ x ∣ + C$

Example 2:

Solve

$(x^{2} + y^{2}) d y = 2 x y d x$

Solutions:

$\frac{d y}{d x} = \frac{2 x y}{x ^{2} + y ^{2}}$

Put y=vx:

$v + x \frac{d v}{d x} = \frac{2 v}{1 + v ^{2}}$

$x \frac{d v}{d x} = \frac{2 v}{1 + v ^{2}} - v$

$x \frac{d v}{d x} = \frac{2 v - v ( 1 + v ^{2} )}{1 + v ^{2}} = \frac{v - v ^{3}}{1 + v ^{2}}$

$\frac{1 + v ^{2}}{v ( 1 - v ^{2} )} d v = \frac{d x}{x}$

6.0Practice Questions

Solve: $\frac{d y}{d x} = \frac{x ^{2} + y ^{2}}{2 x y}$
Solve: $(x^{2} - y^{2}) d y = 2 x y d x$
Show and solve: $\frac{d y}{d x} = \frac{x + y}{x - y}$
Solve: $\frac{d y}{d x} = \frac{3 x ^{2} + y ^{2}}{2 x y}$
Evaluate the solution for: $\frac{d y}{d x} = \frac{x ^{2} - x y}{y ^{2} - x y}$

Frequently Asked Questions

What is the difference between a homogeneous and a non-homogeneous differential equation?

What is the solution form of a homogeneous differential equation?

Can higher-order differential equations be homogeneous?

Are homogeneous differential equations always linear?

What are some common applications of homogeneous differential equations?

Join ALLEN!

Homogeneous Differential Equations: Definition, Formula & Examples

1.0What is a Homogeneous Differential Equation?

Homogeneous Differential Equations Definition

General Form

Condition for Homogeneity

2.0Homogeneous Differential Equations Formula

3.0Steps to Solve Homogeneous Differential Equation

Substitution Method (y=vx)

Derivation of Formula

4.0How to Solve a Homogeneous Differential Equation?

5.0Homogeneous Differential Equations Examples (Solved)

Example 1:

Example 2:

6.0Practice Questions

Table of Contents

Frequently Asked Questions

What is the difference between a homogeneous and a non-homogeneous differential equation?

What is the solution form of a homogeneous differential equation?

Can higher-order differential equations be homogeneous?

Are homogeneous differential equations always linear?

What are some common applications of homogeneous differential equations?

Join ALLEN!

Homogeneous Differential Equations: Definition, Formula & Examples

1.0What is a Homogeneous Differential Equation?

Homogeneous Differential Equations Definition

General Form

Condition for Homogeneity

2.0Homogeneous Differential Equations Formula

3.0Steps to Solve Homogeneous Differential Equation

Substitution Method (y=vx)

Derivation of Formula

4.0How to Solve a Homogeneous Differential Equation?

5.0Homogeneous Differential Equations Examples (Solved)

Example 1:

Example 2:

6.0Practice Questions

Table of Contents