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Homogeneous Differential Equation

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What Is Homogeneous Differential Equation?

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:


dydx = f (yx)


Thus, a differential equation of the first order and of the first degree is homogeneous when the value of dydx is a function of yx. For example, we consider the differential equation:   

                          (x2 + y2) dy - xy dx = 0

 Now,

                           (x2 + y2) dy - xy dx = 0 or,   (x2 + y2) dy - xy dx   

             

 

or,  dydx = xyx2+y2 = yx1+(yx)2 = function of yx


Therefore, the equation   (x2 + y2) dy - xy dx = 0 is a homogeneous equation. On the contrary the differential equation  (x2 + y2) dy - xy2  dx = 0 is not a homogeneous equation since in this case, the value of dydx is not a function of yx

Homogeneous Differential Equation Examples

Example 1) Solve (x2 - xy) dy = (xy + y2)dx 



Solution 1) We have (x2 - xy) dy = (xy + y2)dx ... (1)

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


From (1), we have dydx = xy+y2x2xy.... (2)


Now put y = vx, then dydx = v + x. dydx


From (2), v + x.dydx = x.vx+v2x2x2x.vx = v+v21v



Or,  x dydx = v+v21v - v = v+v2v+v21v = 2v21v

 


Or, 1v2v2 dv = dydx or, dxx = 12 (1v21v)dv


Integrating, log x = 12 (1vlogv) + 12log C



Or, 2 Log x = -  1v - logv + log C or, log x2 + log v - log C = - 1v


OR, Log (vx2C) = - xy  [y = vx] or, vx2C e xy, or, xy = Ce - xy


Which is the required general solution of homogeneous equation examples?

               

Example 2) Solve:  (x2 + y2) dx - 2xy = 0, given that y = 0, when x = 1.


Solution 2)  We have  (x2 + y2) dx - 2xy dy = 0 or, dydx = x2+y22xy … (1)


Put y = vx; then dydx = v + xdvdx 



From, (1), v + x dydx = x2+y2x22x2v = 1+v22v


Or,   2v1v2. dv = dxx or, - (2v1v2)dv = dxx


Integrating - log(1 - v2) = log |x| - log C


Or, Log |1v2| = - log |x| + log C


Or, log Log |1v2| x = log C or, ( 1 - v2)x = C


Or,  (1y2x2) x = C,     or, x2 - y2 = Cx …(2)


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

Hence from (2), the required solution is x2 - y2 = x


Example 3) Solve: {x+ycos(yx)}dx = x cos (yx)


Solution 3) {x+ycosyx} dx = x cos xcos (yx)dy


OR, dxx  = x+ycos(vx)xcos(vx)   put y = vx; then dxdx = v + x dvdx.............(1)



From (1), v + xdvdx = x+vxcos(vxx)xcos(vxx) = 1+vcosvcosv


Or, v + x dvdx = sec v + v, or, xdvdx = sec v, or, cos v. dv = dxx


Integrating both sides, we get cos v dv = dxx + C


OR, sin v = log |x| + C, or, sin yx = 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


                                                       (x2+y2)2xy


Solution 4) for any curve y = f(x), the slope at any point (x,y) is dydx


dydx = x2+y22xy........(1)


Which is a homogeneous differential equation of first order? 


Put y = vx; then dydx = v + xdvdx


From (1), we get v + x  dvdx = x2+y2x22x.vx = 1+v22v  


Or,  xdydx = 1+v22v - v = 1v22v or, dxx   = 2v1v2 dv = 2vdv1v2


Integrating both sides, we get

 

log |x| = -   2vdv1v2 + lof C = -log |1v2| + log C


Or,   log |x| = log |C1v2| = log |Cx2x2y2| or, cx2x2y2 = x, or, x2 - y2 = Cx……….(2)


Where C is an arbitrary constant. 

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

Therefore, 12 - 02 = C. 1, or C = 1.

Hence, from (2), the required equation of the curve is x2 - y2 = x.

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FAQs on Homogeneous Differential Equation

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.