Homogeneous differential equation
A first order differential equation is homogeneous when it can be written in form
$$\frac{dy}{dx}=F(\frac{y}{x})$$
We can solve it using separation of variables but first we make a new variables
$$v=\frac{y}{x}$$
$$v=\frac{y}{x}\;\text{and}\;y=vx$$
$$\frac{dy}{dx}=\frac{d(vx)}{dx}$$
$$v\frac{dx}{dx}+x\frac{dx}{dx}$$
using product rule
Simplified as
$$\frac{dy}{dx}=v+x\frac{dx}{dx}$$
By use of
$$y=vx$$
and
$$\frac{dy}{dx}=v+x$$
we can solve the differential equation. Example will show how it is all done
Example
Solve
$$\frac{dy}{dx}=\frac{x^2+y^2}{xy}$$
Can we make a form like F(y/x)?
Now, we have a function (y/x)
So,
By Separation of variable:
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Step#1:
$$\frac{x^2+y^2}{xy}$$
-
Step#2: Separated term
$$\frac{x^2}{xy}+\frac{y^2}{xy}$$
-
Step#3: Simplify:
$$\frac{x}{y}+\frac{y}{x}$$
-
Step$4:bReciprocal of first term
$$(\frac{x}{y})^{-1}+\frac{y}{x}$$
-
Step#5: Start with
$$(\frac{x}{y})^{-1}+\frac{y}{x}$$
$$y=vx$$
and
$$\frac{dx}{dx}=v+x\frac{dx}{dx}=v^{-1}+v$$
-
Step#6: Subtract from v on both side
$$x\frac{dx}{dx}=v^{-1}$$
-
Step#7: Separate the variable
$$vdv=\frac{1}{x} dx$$
-
Step$8: Use integral on both sides
$$\int vdx=\int \frac{1}{x}dx$$
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Step#9: Perform Integration
$$\frac{v^2}{2}=ln(x)+c$$
-
Step#10: Put
$$c=ln(m)$$
$$\frac{v^2}{2}=ln(x)+ln(m)$$
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Step#11: together
$$\frac{v^2}{2}=ln(m)$$
-
Step#12: Now find v and then take square root on both sides
$$v=\pm\sqrt{2ln(xm)}$$
Now substitute
$$v=\frac{y}{x}$$
$$\frac{y}{x}=\pm\sqrt{2ln(xm)}$$
Now simplify as
$${y}=\pm\;x\sqrt{2ln(xm)}$$