Solution of Differential Equations of the Homogeneous Form

Differential equations (DEs) are mathematical equations that relate some function with its derivatives. When it comes to solving differential equations, one common type is the homogeneous differential equation. In this context, "homogeneous" refers to the zero on the right-hand side of the equation when it is written in standard form.

Homogeneous Differential Equations

A first-order ordinary differential equation is said to be homogeneous if it can be written in the form:

$$ \frac{dy}{dx} = f\left(\frac{y}{x}\right) $$

where ( f ) is a function of the single variable ( \frac{y}{x} ). This form suggests that the change in ( y ) with respect to ( x ) depends only on the ratio ( \frac{y}{x} ), not on ( x ) and ( y ) independently.

Solving Homogeneous Differential Equations

To solve a homogeneous differential equation, we use the substitution:

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

This implies that ( y = vx ) and by differentiating both sides with respect to ( x ), we get:

$$ \frac{dy}{dx} = v + x\frac{dv}{dx} $$

Substituting ( v ) and ( \frac{dy}{dx} ) back into the original differential equation gives us an equation in terms of ( v ) and ( x ):

$$ v + x\frac{dv}{dx} = f(v) $$

This is now a separable differential equation, which can be solved by separating the variables ( v ) and ( x ) and integrating both sides.

Steps to Solve Homogeneous Differential Equations

1. Make the substitution ( v = \frac{y}{x} ).
2. Express ( y ) in terms of ( v ) and ( x ): ( y = vx ).
3. Differentiate ( y ) with respect to ( x ) to find ( \frac{dy}{dx} ).
4. Substitute ( v ) and ( \frac{dy}{dx} ) into the original DE to obtain an equation in ( v ) and ( x ).
5. Separate the variables and integrate both sides.
6. Solve for ( v ) as a function of ( x ).
7. Substitute back ( v = \frac{y}{x} ) to find ( y ) as a function of ( x ).

Example

Solve the homogeneous differential equation:

$$ \frac{dy}{dx} = \frac{x + y}{x} $$

Solution

1. Let ( v = \frac{y}{x} ), then ( y = vx ).
2. Differentiate ( y ) with respect to ( x ):

$$ \frac{dy}{dx} = v + x\frac{dv}{dx} $$

1. Substitute ( v ) and ( \frac{dy}{dx} ) into the original DE:

$$ v + x\frac{dv}{dx} = 1 + v $$

1. Simplify and separate variables:

$$ x\frac{dv}{dx} = 1 $$

1. Integrate both sides:

$$ \int x\frac{dv}{dx} dx = \int 1 dx $$

$$ v \cdot x = x + C $$

1. Solve for ( v ):

$$ v = 1 + \frac{C}{x} $$

1. Substitute back ( v = \frac{y}{x} ) to find ( y ):

$$ y = x + C $$

Differences and Important Points

Aspect	Homogeneous DE	Non-Homogeneous DE
Definition	DE with zero on the right-hand side when written in standard form	DE with a non-zero term on the right-hand side
General Form	( \frac{dy}{dx} = f\left(\frac{y}{x}\right) )	( \frac{dy}{dx} = g(x, y) + h(x) )
Solution Method	Substitution ( v = \frac{y}{x} )	Various methods (e.g., undetermined coefficients, variation of parameters)
Example	( \frac{dy}{dx} = \frac{x + y}{x} )	( \frac{dy}{dx} = x + y + e^x )

In summary, homogeneous differential equations can be solved by using a substitution that simplifies the equation into a separable form. This method relies on the fact that the change in the dependent variable is a function of the ratio of the dependent to the independent variable. Once the equation is separated and integrated, the solution can be expressed in terms of the original variables.