Solution of Differential Equations Reducible to the Homogeneous Form

Differential equations (DEs) are mathematical equations that relate functions with their derivatives. They are used to model various phenomena in physics, engineering, biology, economics, and more. One important class of differential equations is the first-order ordinary differential equations (ODEs) that can be reduced to a homogeneous form.

Homogeneous Differential Equations

A first-order ODE is called homogeneous if it can be written in the form:

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

where $F$ is a function of a single variable. The key characteristic of a homogeneous function is that it exhibits multiplicative scaling behavior, meaning that for any real number $\lambda$, the function satisfies:

$$ F(\lambda y) = \lambda^n F(y) $$

where $n$ is the degree of homogeneity.

Reducing Non-Homogeneous to Homogeneous Form

A non-homogeneous first-order ODE can often be reduced to a homogeneous form through an appropriate substitution. The general form of a non-homogeneous first-order ODE is:

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

To reduce this to a homogeneous form, we look for a substitution that will eliminate the explicit dependence on $x$ and $y$ separately, leaving a function that depends only on the ratio $y/x$ or $x/y$.

Common Substitutions

1. For equations of the form $\frac{dy}{dx} = f\left(\frac{ax + by + c}{dx + ey + f}\right)$:

If $c = f = 0$, the equation is already homogeneous. Otherwise, we can use the substitution $x = x' - \frac{c}{a}$ and $y = y' - \frac{f}{d}$ (assuming $a$ and $d$ are not zero) to eliminate the constants and reduce the equation to a homogeneous form.

1. For equations of the form $\frac{dy}{dx} = g(ax + by) + h(cx + dy)$:

If $g$ and $h$ are homogeneous functions of the same degree, the equation can be made homogeneous by dividing through by one of the terms (usually the one involving $x$), resulting in a function of $y/x$ or $x/y$.

Steps to Solve

1. Identify the substitution that will reduce the equation to a homogeneous form.
2. Perform the substitution and simplify the equation.
3. Solve the resulting homogeneous equation.
4. Reverse the substitution to find the solution to the original non-homogeneous equation.

Table of Differences and Important Points

Feature	Homogeneous DE	Non-Homogeneous DE
Form	$\frac{dy}{dx} = F\left(\frac{y}{x}\right)$	$\frac{dy}{dx} = f(x, y)$
Solution Method	Direct integration after substitution	Requires reduction to homogeneous form
Substitution	Not required	Required to eliminate explicit $x$ and $y$ dependence
Example	$\frac{dy}{dx} = \frac{y - x}{y + x}$	$\frac{dy}{dx} = \frac{2x + 3y - 5}{x + y - 2}$

Examples

Example 1: Homogeneous DE

Consider the homogeneous differential equation:

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

Substitute $v = \frac{y}{x}$, which gives $y = vx$ and $\frac{dy}{dx} = v + x\frac{dv}{dx}$. The equation becomes:

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

Solve this equation for $v(x)$, and then substitute back to find $y(x)$.

Example 2: Non-Homogeneous DE Reducible to Homogeneous Form

Consider the non-homogeneous differential equation:

$$ \frac{dy}{dx} = \frac{2x + 3y - 5}{x + y - 2} $$

We can use the substitution $x = x' + 2$ and $y = y' + 5$ to eliminate the constants. The equation becomes:

$$ \frac{dy'}{dx'} = \frac{2x' + 3y'}{x' + y'} $$

This is now a homogeneous equation. Solve for $y'(x')$, and then reverse the substitution to find $y(x)$.

Conclusion

Reducing a non-homogeneous differential equation to a homogeneous form is a powerful technique for finding solutions to a wide range of problems. By using appropriate substitutions, we can transform the problem into one that is more easily solvable, and then translate the solution back to the original variables. Understanding this method is essential for anyone studying differential equations, as it provides a pathway to solving complex problems that might otherwise seem intractable.