derive a differential equation of order 2

Question

Please derive a differential equation of order 2 having general solution

: x^2/c1 + y^2/c2 =1

c1 and c2 is constant.

Please write the details.

Answer 1

Since there are two constants we will have to differentiate the given equation twice in order to eliminate the constants $c_1$ and $c_2$. First derivative gives us: $$x/c_1+yy'/c_2=0\Rightarrow yy'=-(c_2/c_1)x.$$ Second derivative yields: $$(y')^2+yy''=-(c_2/c_1).$$ On dividing both equationswe get: $$\frac{(y')^2+yy''}{yy'}=\frac{1}{x}\Rightarrow x(y')^2+xyy''-yy'=0.$$ which is the required differential equation.