The first question that comes to our mind is what is a homogeneous equation? Well, let us start with the basics. We know that the differential equation of the first order and of the first degree can be expressed in the form Mdx + Ndy = 0, where M and N are both functions of x and y or constants. In particular, if M and N are both homogeneous functions of the same degree in x and y, then the equation is said to be a homogeneous equation. Such an equation can be expressed in the following form:

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

Thus, a differential equation of the first order and of the first degree is homogeneous when the value of \[\frac{dy}{dx}\] is a function of \[\frac{y}{x}\]. For example, we consider the differential equation:

(\[x^{2}\] + \[y^{2}\]) dy - xy dx = 0

Now,

(\[x^{2}\] + \[y^{2}\]) dy - xy dx = 0 or, (\[x^{2}\] + \[y^{2}\]) dy - xy dx

or, \[\frac{dy}{dx}\] = \[\frac{xy}{x^{2} + y^{2}}\] = \[\frac{\frac{y}{x}}{1 + \left ( \frac{y}{x}\right )^{2}}\] = function of \[\frac{y}{x}\]

Therefore, the equation (\[x^{2}\] + \[y^{2}\]) dy - xy dx = 0 is a homogeneous equation. On the contrary the differential equation (\[x^{2}\] + \[y^{2}\]) dy - \[xy^{2}\] dx = 0 is not a homogeneous equation since in this case, the value of \[\frac{dy}{dx}\] is not a function of \[\frac{y}{x}\]

Example 1) Solve (\[x^{2}\] - xy) dy = (xy + \[y^{2}\])dx

Solution 1) We have (\[x^{2}\] - xy) dy = (xy + \[y^{2}\])dx ... (1)

The differential equation (1) is a homogeneous equation in x and y.

From (1), we have \[\frac{dy}{dx}\] = \[\frac{xy + y^{2}}{x^{2} - xy}\].... (2)

Now put y = vx, then \[\frac{dy}{dx}\] = v + x. \[\frac{dy}{dx}\]

From (2), v + x.\[\frac{dy}{dx}\] = \[\frac{x.vx + v^{2x^{2}}}{x^{2} -x.vx}\] = \[\frac{v +v^{2}}{1-v}\]

Or, x \[\frac{dy}{dx}\] = \[\frac{v + v^{2}}{1-v}\] - v = \[\frac{v + v^{2} - v + v^{2}}{1-v}\] = \[\frac{2v^{2}}{1-v}\]

Or, \[\frac{1-v}{2v^{2}}\] dv = \[\frac{dy}{dx}\] or, \[\frac{dx}{x}\] = \[\frac{1}{2}\] \[\left ( \frac{1}{v^{2}} - \frac{1}{v}\right )\]dv

Integrating, log x = \[\frac{1}{2}\] \[\left ( - \frac{1}{v} - logv \right )\] + \[\frac{1}{2}\]log C

Or, 2 Log x = - \[\frac{1}{v}\] - logv + log C or, log \[x^{2}\] + log v - log C = - \[\frac{1}{v}\]

OR, Log \[\left ( \frac{vx^{2}}{C} \right )\] = - \[\frac{x}{y}\] [y = vx] or, \[\frac{vx^{2}}{C}\] e \[\frac{x}{y}\], or, xy = Ce - \[\frac{x}{y}\]

Which is the required general solution of homogeneous equation examples?

Example 2) Solve: (\[x^{2}\] + \[y^{2}\]) dx - 2xy = 0, given that y = 0, when x = 1.

Solution 2) We have (\[x^{2}\] + \[y^{2}\]) dx - 2xy dy = 0 or, \[\frac{dy}{dx}\] = \[\frac{x^{2} + y^{2}}{2xy}\] … (1)

Put y = vx; then \[\frac{dy}{dx}\] = v + x\[\frac{dv}{dx}\]

From, (1), v + x \[\frac{dy}{dx}\] = \[\frac{x^{2} + y^{2}x^{2}}{2x^{2}v}\] = \[\frac{1 + v^{2}}{2v}\]

Or, \[\frac{2v}{1-v^{2}}\]. dv = \[\frac{dx}{x}\] or, - \[\left ( \frac{-2v}{1-v^{2}} \right )\]dv = \[\frac{dx}{x}\]

Integrating - log(1 - \[v^{2}\]) = log |x| - log C

Or, Log \[\left | 1 - v^{2} \right |\] = - log |x| + log C

Or, log Log \[\left | 1 - v^{2} \right |\] x = log C or, ( 1 - \[v^{2}\])x = C

Or, \[\left ( 1 - \frac{y^{2}}{x^{2}} \right )\] x = C, or, \[x^{2}\] - \[y^{2}\] = Cx …(2)

Given y = 0 when x = 1; therefore, from (2), 1 = C

Hence from (2), the required solution is \[x^{2}\] - \[y^{2}\] = x

Example 3) Solve: \[\left \{ x + y cos\left ( \frac{y}{x} \right ) \right \}\]dx = x cos \[\left ( \frac{y}{x} \right )\]

Solution 3) \[\left \{ x + y cos\frac{y}{x} \right \}\] dx = x cos xcos \[\left ( \frac{y}{x} \right )\]dy

OR, \[\frac{dx}{x}\] = \[\frac{x +y cos\left ( \frac{v}{x} \right )}{x cos\left ( \frac{v}{x} \right )}\] put y = vx; then \[\frac{dx}{dx}\] = v + x \[\frac{dv}{dx}\].............(1)

From (1), v + x\[\frac{dv}{dx}\] = \[\frac{x + vx cos \left ( \frac{vx}{x} \right )}{ x cos\left ( \frac{vx}{x} \right )}\] = \[\frac{1+v cosv}{cosv}\]

Or, v + x \[\frac{dv}{dx}\] = sec v + v, or, x\[\frac{dv}{dx}\] = sec v, or, cos v. dv = \[\frac{dx}{x}\]

Integrating both sides, we get \[\int\] cos v dv = \[\int\] \[\frac{dx}{x}\] + C

OR, sin v = log |x| + C, or, sin \[\frac{y}{x}\] = log|x| + C

Which is the required solution of (1) for the homogeneous equation examples?

Example 4) Find the equation to the curve through (1,0) for which the slope at any point (x, y) is

\[\frac{(x^{2} + y^{2})}{2xy}\]

Solution 4) for any curve y = f(x), the slope at any point (x,y) is \[\frac{dy}{dx}\]

\[\frac{dy}{dx}\] = \[\frac{x^{2} + y^{2}}{2xy}\]........(1)

Which is a homogeneous differential equation of first order?

Put y = vx; then \[\frac{dy}{dx}\] = v + x\[\frac{dv}{dx}\]

From (1), we get v + x \[\frac{dv}{dx}\] = \[\frac{x^{2}+y^{2}x^{2}}{2x.vx}\] = \[\frac{1 + v^{2}}{2v}\]

Or, x\[\frac{dy}{dx}\] = \[\frac{1+v^{2}}{2v}\] - v = \[\frac{1- v^{2}}{2v}\] or, \[\frac{dx}{x}\] = \[\frac{2v}{1-v^{2}}\] dv = \[\frac{-2vdv}{1-v^{2}}\]

Integrating both sides, we get

log |x| = - \[\int\] \[\frac{-2vdv}{1-v^{2}}\] + lof C = -log \[\left |1 - v^{2} \right |\] + log C

Or, log |x| = log \[\left | \frac{C}{1-v^{2}} \right |\] = log \[\left | \frac{Cx^{2}}{x^{2}-y^{2}} \right |\] or, \[\frac{cx^{2}}{x^{2}-y^{2}}\] = x, or, \[x^{2}\] - \[y^{2}\] = Cx……….(2)

Where C is an arbitrary constant.

Since this curve passes through the point (1,0);

Therefore, \[1^{2}\] - \[0^{2}\] = C. 1, or C = 1.

Hence, from (2), the required equation of the curve is \[x^{2}\] - \[y^{2}\] = x.

FAQ (Frequently Asked Questions)

1. What differential equation means?

Many important problems in Physical Science, Engineering, and, Social Science lead to equations involving derivatives or differentials when they are expressed in mathematical terms. Such equations are called differential equations.

Thus an equation involving a derivative or differentials with or without the independent and dependent variable is called a differential equation. The order of a differential equation is the order of the highest order derivative or differential appearing in the equation whereas the degree of a differential equation is the degree of the highest order derivative or differential when the derivatives are free from radicals and negative indices.