The general solution of the differential equation dy/dx=x/y is given by y2 = x2 + C where C is an integral constant. Here we will learn how to find the solution of dy/dx=x/y.
General Solution of dy/dx=x/y
Question: Find the general solution of dy/dx=x/y.
The given differential equation is
Separating the variables x and y, we get that
y dy = x dx
Integrating both sides, we get that
∫y dy = ∫x dx +C’ where C’ is a constant of integration.
⇒ y2/2 = x2/2 + C/2 where C’=C/2
⇒ y2 = x2 + C
So the general solution of dy/dx=x/y is equal to y2 = x2 + C where C is an arbitrary constant.
Particular Solution of dy/dx=x/y
Question: Find the particular solution of dy/dx = x/y, y(0)=0.
From above, the solution of dy/dx=x/y is equal to y2 = x2 + C.
Given that y(0)=0, so y=0 when x=0.
Thus from y2 = x2 + C we get that
02 = 02 + C
So the particular solution of dy/dx=x/y when y(0)=0 is equal to y2 = x2, that is y=±√x.
Q: What is the general solution of dy/dx=x/y?
Answer: The general solution of dy/dx=x/y is y2 = x2 + C where C is an integration constant.