Question
Question: Solve the following: \(\left( {{x}^{2}}+{{y}^{2}} \right)dx+\left( 2xy \right)dy=0\)...
Solve the following:
(x2+y2)dx+(2xy)dy=0
Explanation
Solution
Hint: We will be using the concepts of differential equations to solve the problem. We will first test the given differential equation for homogeneity then we will use the method of solving homogeneous differential equation to solve the problem in which we make the substitution y=υx to convert the differential equation in one variable v.
Complete step-by-step answer:
Now, we have been given that,
(x2+y2)dx+(2xy)dy=0−2xyx2+y2=dxdy
Now, we will check the given differential for homogeneity for that we put F(x,y)=dxdy and will find F(λx,λy).
F(x,y)=2xy−(x2+y2)
So, we have,