Q The curve satisfying the differential equation , (x2-y2) dx +2xy dy = 0 and passing through the - Maths - Differential Equations

Question

Q The curve satisfying the differential equation ,
(x2-y2) dx +2xy dy = 0 and passing through
the point (1,1) is : (a) an ellipse (b) a hyperbola (c) a circle of
radius one (d) a circle of radius two

Neha Sethi · Accepted Answer

Dear student
x2-y2dx+2xydy=0⇒x2-y2dx=-2xy dy⇒dydx=-x2-y22xy⇒dydx=y2-x22xy  
1which is a homogeneous  differential equation
Let y=vx⇒dydx=v+xdvdx  
2From 1 and 2, we get⇒v+xdvdx  =v2x2-x22vx2⇒v+xdvdx  =v2-12v⇒xdvdx=v2-12v-v⇒xdvdx=v2-1-2v22v⇒xdvdx=-1-v22v⇒xdvdx=-1+v22v⇒2v1+v2dv=-dxxIntegrating both sides, we get⇒∫2v1+v2dv=-∫dxx⇒log1+v2=-logx+C⇒log1+v2=-logx+logC⇒log1+v2+logx=logC⇒logx1+v2=logC⇒x1+v2=C⇒x1+yx22=C⇒x2+y2x2x=C⇒x2+y2=Cx  Given x=1 and y=1⇒12+12=C⇒C=2So, x2+y2=2x  ⇒ x2+y2-2x =0⇒x2-2x+12-12+y2=0⇒x-12+y2=1which represents a circle with centre 1,0 and radius 1So, correct option is cNote: logm+logn=logm
n
Regards

Q. The curve satisfying the differential equation , (x2-y2) dx +2xy dy = 0 and passing through the point (1,1) is : (a) an ellipse (b) a hyperbola (c) a circle of radius one (d) a circle of radius two

Q. The curve satisfying the differential equation , (x²-y²) dx +2xy dy = 0 and passing through the point (1,1) is : (a) an ellipse (b) a hyperbola (c) a circle of radius one (d) a circle of radius two