Question
The curves satisfying the differential equation (1−x2)y1+xy=ax are
Answer: Option A
:
A
The given equation is linear DE and can be written as
dydx+x1−x2y=ax1−x2
Its integrating factor is e∫x1−x2dx=e−(12)log(1−x2)=1√1−x2−1<x<1 and ifx2>1then 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 -1 < x < 1 the given equation represents an ellipse. If x2>1then the solution is of the form −(y−a)2+C2x2=C2which represents a hyperbola.
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:
A
The given equation is linear DE and can be written as
dydx+x1−x2y=ax1−x2
Its integrating factor is e∫x1−x2dx=e−(12)log(1−x2)=1√1−x2−1<x<1 and ifx2>1then 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 -1 < x < 1 the given equation represents an ellipse. If x2>1then the solution is of the form −(y−a)2+C2x2=C2which represents a hyperbola.
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