Question

Which of the following differential equations can be solved using variable separable method.
$1.(x^2+y^2)dx-2xydy=0$
$2.(x^2+\sin x)dx+\left(\sin y\right)dy=0$

• A. Only 1
• B. Only 2
• C. Both 1 and 2
• D. None of the above

Answer 1

The correct option is B Only 2

Look carefully at first equation. Do you think that we can separate all $x$'s at one side and all $y$'s on another side.
No it’s not possible for this equation. There will always be a $y$ associated with a $dx$ or a $x$ associated with $dy$. So the variables can’t be separated. So this cannot be solved by variable separable method.
Second equation: $(x^2+\sin x)dx+\left(\sin y\right)dy=0$
This already has all $x$ terms separated from all $y$ terms. So this can be solved using variable separable method.