Differential Equation
Formation of Differential Equation
Question: Find the differential equation that describes the family of circles passing through the origin.
Solution: The general form of the equation of the circle is
(x−h)2+(y−k)2=√(h2+k2)2⟹x2−2xh+h2+y2−2yk+k2=h2+k2⟹x2−2xh+y2−2ky=0(x−h)2+(y−k)2=√(h2+k2)2⟹x2−2xh+h2+y2−2yk+k2=h2+k2⟹x2−2xh+y2−2ky=0(1.1)
Using differentiation twice we get,
2x−2h+2yy′−2ky′=0⟹x−h+yy′−ky′=0and 1−0+yy′+y′y′−ky″=0⟹1+yy′+(y′)2−ky″=0
From the equation (1.1) we get,
2xh=x2+y2−2kyh=(x2+y2−2ky)2x
From equation (1.2) and (1.4) we get,
x−x2+y2−2ky2x+yy′−ky′=0⟹2x2−x2−y2+2ky+2xyy′−2xky′=0⟹2k(y−xy′)=−x2+y2−2xyy′⟹ k=−x2+y2−2xyy′2(y−xy′)⟹ k=x2−y2+2xyy′2(xy′−y)
Again from (1.3) and (1.5) we get,
1+yy″+(y′)2−x2−y2+2xyy′2(xy′−y)y″=0⟹2[1+(y′)2](xy′−y)+2yy″(xy′−y)−(x2−y2+2xyy′)y″=0⟹2[1+(y′)2](xy′−y)+2xyy′y″−2y2y″−x2y″+y2−2xyy′y″=0
Which is the required differential equation.
Question: Find the differential equation of the family y=2ce2x1+ce2x
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