Differential Equation
Formation of Differential Equation
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Question: Form the differential equation of the curve represented by y2−2ay+x2=a2y2−2ay+x2=a2. where a is an arbitrary constant.
Solution: The given equation is,
y2−2ay+x2=a2
Differentiating both sides of (1.1) with respect to x we get,
2ydydx−2adydx+2x=0⟹ydydx+x=adydx⟹y+xdxdy=a
Substituting this value of a in (1.1) we have,
y2−2h(y+xdxdy)+x2=(y+xdxdy)2⟹y2−2y2−2xydxdy+x2=y2+2xydxdy+x2(dxdy)2⟹2y2+4xydxdy+x2(dxdy)2−x2=0⟹2y2(dydx)2+4xydydx+x2−x2(dydx)2=0
Solution: We know, the equation of all straight lines passing through the origin is,
y=mx
Differentiating (2.1).
dydx=m
eliminating m between (2.1) and (2.2)
i.e by putting m=dydx in y=mx
we get, y=xdydx
Which is the required differential equation.
Solution: The given equation is,
  y=ae2x+be−3x+cex
Differentiating both sides of (3.1) with respect o x, we get
dydx=2ae2x−3be−3x+cex
Subtracting (3.1) from (3.2) we get,
dydx−y=ae2x−4be−3x
Differentiating (3.3) we get,
d2ydx2−dydx=3ae2x+12be−3x
Adding 3 times of (3.3) to (3.4) we have,
(d2ydx2−dydx)+d(dydx−y)=(2ae2x+12be−3x)+3(ae2x−4be−3x)d2ydx2+2dydx−3y=5ae2x
Differentiating both sides of (3.5) with respect to x we get,
d3ydx3+2d2ydx2−dydx=10ae2x
From (3.5) and (3.6) eliminating a we get,
  d3ydx3+2d2ydx2−3dydx=2(d2ydx2+2dydx−3y)⟹ d3ydx3−7dydx+6y=0
Solution: Given equation is,
xy=Aex+Be−x+x2
differentiating with respect to x we get,
xdydx+y=Aex−Be−x+2x
Differentiating again we get,
xd2ydx2+dydx+dydx=Aex+Be−x+2xd2ydx2+2dydx=xy−x2+2
Thus the two variables A and B have been eliminated and we get the differential equation of the 2nd order as,
xd2ydx2+2dydx=xy−x2+2
Solution: The given equation is,
y=ex(Acosx+Bsinx)
Differentiating with respect to x, we get
dydx=ex(−Asinx+Bcosx)+ex(Acosx+Bsinx)dydx=ex(−Asinx+Bcosx)+y
Again Differentiating both sides with respect to x, we get
d2ydx2=ex(−Acosx−Bsinx)+ex(−Asinx+Bcosx)+dydx=−ex(Acosx+Bsinx)+(dydx−y)+dydx−y+(dydx−y)+dydx∴ d2ydx2−2dydx+2y=0
Solution: The given equation is,
y=cx3c=yx3
Now
y=cx3dydx=3cx2dydx=3yx3x2dydx=3yxxdydx=3yxdydx−3y=0
which is the required differential equation.
1. Show that the differential equation of the family of the circle touches the x-axis at the origin is
(x2−y2) dy−2xy dx=0
2. Find the Differential Equation of the family of curves
y=ex(Acosx+Bsinx)
where A and B are arbitrary constants.
3. Discuss the degree and order of the differential equations with examples.
4. Find the differential equation of all circles of radius a.
5. Form a differential equation from
y=Acosax+Bsinax
, where A and B arbitrary constants and a is a fixed constant.
6. Find the differential equation of all circles which pass through the origin and whose centers are on the x-axis.
7. briefly discuss differential equation with its degree and order. Form an ordinary differential equation from
y=ex(acosx+bsinx)
, where a and b parameter.
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