Homogeneous Differential Equations 

A homogeneous differential equation is an equation in which all terms involve the dependent variable and its derivatives, and the equation equals zero. These equations are crucial in fields like physics and engineering, where systems in equilibrium or balance are modeled. Homogeneous differential equations can be classified as first-order or second-order, and solving them often involves techniques like substitution, separation of variables, or solving characteristic equations. Their solutions help describe dynamic systems in various scientific applications.

1.0Homogeneous Differential Equation Meaning

A function f(x, y) is called a homogeneous function of degree n if the substitution
x = λx, y = λy leads to:

$f(\lambda x,\lambda y)={\lambda }^{n}f(x,y)$

Here, λ > 0 and n is a real number indicating the degree of homogeneity.

2.0First-Order Homogeneous Differential Equation

A differential equation of the form:

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

is homogeneous if both f(x, y) and g(x, y) are homogeneous functions of the same degree.

Method of Solving

Substitute:

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

This transforms the equation into a separable form in terms of v and x.

Sometimes, substituting x = vy or using polar coordinates helps.

3.0Equations Reducible to Homogeneous Form

A differential equation of the form:

 $\frac{dy}{dx}=\frac{a_{1}x+b_{1}y+c_1}{a_{2}x+b_{2}y+c_2}$

can be transformed into homogeneous form using the substitution:

x = u + h, y = v + k  

Find constants h and k by solving:

$a_{1}h+b_{1}k+c_1=0,a_{2}h+b_{2}k+c_2=0$

4.02nd-Order Homogeneous Differential Equation

A second-order homogeneous differential equation is of the form:

$a\frac{d^{2}y}{d{x}^2}+b\frac{dy}{dx}+cy=0$

Solutions typically involve finding the roots of the auxiliary equation: 

$a{r}^2+br+c=0$

and solving based on the nature of the roots (real, repeated, complex).

5.0Homogeneous Differential Equation Formula

General Form

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

Substitute y = vx, then

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

Reduce to a separable form and integrate.

6.0General Solution for Homogeneous Differential Equation

Once the differential equation is reduced to a separable form, integrate both sides:

$\int \frac{dv}{G(v)}=\int \frac{dx}{x}$

Finally, substitute back v = y/x (or x/y) to express the general solution.

7.0Solved Examples on Homogeneous Differential Equations

Example 1: Find the degree of homogeneity for the following:

1. $f(x,y)=x^2+y^2$

Solution: 

$$\begin{gathered} f(\lambda x,\lambda y)={\lambda }^2{x}^2+{\lambda }^2{y}^2={\lambda }^2(x^2+y^2) \\ \Rightarrow \text{Degree}=2 \end{gathered}$$

2. $f(x,y)=\frac{x^{3/2}+y^{3/2}}{x+y}$

Solution: 

$$\begin{gathered} f(\lambda x,\lambda y)=\frac{{\lambda }^{1/2}{x}^{1/2}+{\lambda }^{1/2}{y}^{1/2}}{\lambda x+\lambda y} \\ =\frac{{\lambda }^{1/2}(x^{1/2}+y^{1/2})}{\lambda (x+y)}={\lambda }^{-1/2}f(x,y) \\ \Rightarrow \text{Degree}=\frac{1}{2} \end{gathered}$$

3. $f(x,y)=\frac{x}{x+y}+\frac{y}{x+y}=1$

Solution: 

$$\begin{gathered} f(\lambda x,\lambda y)=1={\lambda }^{0}f(x,y) \\ \Rightarrow \text{Degree}=0 \end{gathered}$$

Example 2: Check if the following functions are homogeneous:

1. $f(x,y)=x^2-xy$

Solution: 

$$\begin{gathered} f(\lambda x,\lambda y)={\lambda }^2(x^2-xy)={\lambda }^{2}f(x,y) \\ \Rightarrow \text{Yes, Degree}=2 \end{gathered}$$

2. $f(x,y)=\frac{x^2+y}{x+y^2}$

Solution: 

$f(\lambda x,\lambda y)\ne {\lambda }^{n}f(x,y)$

3. $f(x,y)=\sin (xy)$

Solution: 

$$\begin{gathered} f(\lambda x,\lambda y)=\sin ({\lambda }^{2}xy)\ne {\lambda }^n\sin (xy) \\ \Rightarrow \text{Not Homogeneous} \end{gathered}$$ 

Example 3: Solve:

$(1+2e^{x/y})dx+2e^{x/y}(1-x/y)dy=0$

Solution:

The equation is homogeneous (degree 0).

Substitute

$x=vy\Rightarrow dx=vdy+ydv$

Equation becomes:

$(v+2e^v)dy+y(1+2e^v)dv=0$

Separate and integrate:

$\ln y+\ln (v+2e^v)=\ln C\Rightarrow x+2y{e}^{x/y}=C$

Example 4: Solve:

$\frac{dy}{dx}=\frac{x+2y+3}{2x+3y+4}$

Solution:

Transform by setting x = X + 1, y = Y – 2 

8.0Practice Questions on Homogeneous Differential Equations 

1. Solve: $\frac{dy}{dx}=\frac{x^2+y^2}{xy}$
2. Determine whether $f(x,y)=\frac{x^2-y^2}{x^2+y^2}$ is homogeneous.
3. Solve: $(x^2+xy)dx+(xy+y^2)dy=0$
4. Find the degree of homogeneity of $f(x,y)=\ln (xy)$
5. Solve: $\frac{dy}{dx}=\frac{3x+2y+1}{4x+5y+6}$