A function f (x, y) is called homogeneous function of degree n if

For Example:

Here

So this is homogeneous function of degree 4.

To solve this type of question we put

or

depending on the question.

After putting this we use the same method we use earlier in the question. And we get differential equation in z and x(or y) and then we solve that differential equation and after solution we put back value of z.

Example: Solve

Solution:

So this is a homogeneous equation. Here we put

y = vx

dy = v dx + x dv

So the differential equation is

Integrate both sides

Now put back value of v = y/x

## DIFFERENTIAL EQUATION REDUCIBLE TO HOMOGENEOUS FORMS

We see in the 3^{rd} point about the homogeneous from of equation. Here we see another equation that can be reducible to homogenous from.

Now we can reduce to homogeneous form. We have seen this type of equation earlier but there ‘aB = bA’ So these a little difference between both.

So to solve this type of differential equation we put

x =X+h and y =Y+k

here h and k are constants and we chose these constants such that

So the new equation is

So here we chose h and k such that they satisfy the equations

Ah+bk+c = 0 and Ah+Bk+C = 0

So the value of h and k are

and we can find these value only if the denominator is NOT zero, hence the above condition.

Now if we find these value h and k then the equation become:

Now here we can use the Homogeneous form solution by putting

Y = VX or X = VY

After this, the rest of the solution remains the same as discussed in the previous section.

Example:

Solution:

Here we can see that ½ is not equal to 2/1. So We use the above method to solve this question.

Let’s put

x = X+h and y = Y+k, and

So the equation is

So here h and k be such that h+2k-5 = 0 and 2h+k-4 = 0

Put k from 2^{nd} to 1^{st}

h+2(4-2h)-5 =0

h+8-4h-5=0

Which gives, h=1 and k=2

So the equation is

Now put Y=VX so dY = VdX + XdV

Equation is

So now we integrate both sides.

Put back V = Y/X

So

Here C = c^{2}

Now put

X = x-1 and Y = y-2

So this is our final answer. Here we use substitutions twice to solve this.

Stay tuned for more.

Click on the links below to read more about Differential Equations:

**How to Solve Differential Equations by Variable Separable Method**

**How to Solve Linear Differential Equations**

**Order, Degree & Solutions of Differential Equations**

*Click on the links below to access the list:*

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All the best!!

Team Gradeup

## Comments

write a commentUriti ManishJan 20, 2018

Uriti ManishJan 20, 2018

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jayesh shahAug 16, 2018

KS VASUDHAAug 17, 2018

Aryan AnandJun 24, 2019

Aryan AnandJun 24, 2019