Question
Question: Show that the differential equation \(\left[ x{{\sin }^{2}}\left( \dfrac{y}{x} \right)-y \right]dx+x...
Show that the differential equation [xsin2(xy)−y]dx+xdy=0 is homogeneous. Find the particular solution of this differential equation, given that y=4π when x=1.
Solution
Homogeneous differential equation can be represented in the form dxdy=f(xy) , so we convert the given equation in the above form to show that the equation given is homogeneous. Then, to find the particular solution of this differential equation we put y=vx and calculate the value and integrate the obtained equation to get the desired solution.
Complete step by step answer:
We have been given the differential equation [xsin2(xy)−y]dx+xdy=0
We have to show that the given equation is homogeneous and find the particular solution of this differential equation.
Now, first of all we need to check that the given equation is homogeneous or not.
We have [xsin2(xy)−y]dx+xdy=0
We divide the whole equation by dx, we get