Solving Linear Differential Equations of First Order
The curves sa...
Question
The curves satisfying the differential equation (1−x2)y1+xy=ax are
A
ellipses and hyperbolas
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B
ellipses and parabola
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C
ellipses and straight lines
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D
circles and ellipses
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Solution
The correct option is A ellipses and hyperbolas The given equation is linear in and can be written as dydx+x1−x2y=ax1−x2
Its integrating factor is e∫x1+−x2dt=e−12log(1−x2)=1√1−x2 if −1<x<1 and if x2>1 then I.F.=1√x2−1 ddx(y1√1−x2)=ax(1−x2)32=−12a−2x(1−x2)32⇒y1√1−x2=a√1−x2+C⇒y=a+C√1−x2⇒(y−a)2=C2(1−x2)⇒(y−a)2+C2x2=C2
Thus if the given equation represents an ellipse. If x2>1 then the solution is of the form −(y−a)2+C2x2=C2 which represents a hyperbola.