IAS PYQs 1
2000
1) Show that 3d2ydx2+4xdydx−8y=0 has an integral which is a polynomial in x. Deduce the general solution.
[10M]
2) Reduce d2ydx2+Pdydx+Qy=R, where, P,Q,R are functions of x, to the normal form. Hence solve d2ydx2−4xdydx+(4x2−1)y=−3ex2sin2x
[10M]
3) Solve the differential equation y=x−2ap+ap2. Find the singular solution and interpret it geometrically.
[10M]
4) Show that (4x+3y+1)dx+(3x+2y+1)dy=0 represents a family of hyperbolas with a common axis and tangent at the vertex.
[10M]
5) Solve xdydx−y=(x−1)(d2ydx2−x+1) by the method of variation of parameters.
[10M]
1999
1) Solve the differential equation xdx+ydyxdy−ydx=(1−x2−y2x2+y2)1/2
[10M]
2) Solve d3ydx3−3d2ydx2+4dydx−2y=ex+cosx
[10M]
3) By the method of variation of parameters, solve the differential equation d2ydx2+a2y=sec(ax).
[10M]
1998
1) Solve the differential equation: xy−dydx=y3e−x2
[10M]
2) Show that the equation: (4x+3y+1)dx+(3x+2y+1)dy=0 represents a family of hyperbolas having as asymptotes the lines x+y=0,2x+y+1=0.
[10M]
3) Solve the differential equation: y=3px+4p2
[10M]
4) Solve the differential equation: d2ydx2−5dydx+6y=e4x(x2+9)
[10M]
5) Solve the differential equation: d2ydx2+2dydx+y=xsinx
[10M]
6) Solve the differential equation: x3d3ydx3+2x2d2ydx2+2y=10(x+1x).
[10M]
1997
1) Solve the initial value problem
dydx=xx2y+y3,y(0)=0[10M]
2) Solve (x2−y2+3x−y)dx+(x2−y2+x−3y)dy=0.
[10M]
3) Assume that a spherical rain drop evaporates at a rate proportional to its surface area. If its radius originally is 3 mm, and one hour later has been reduced to 2 mm. find an expression for the radius of the rain drop at any time.
[10M]
4) Solve d4ydx4+6d3ydx3+11d2ydx26dydx=20e−2xsinx
[10M]
5) Make use of the transformation y(x)=u(x)secx to obtain the solution of
y′′−2y′tanx+5y=0,y′(0)=0,y′(0)=√6
[10M]
6) Solve (1+2x)2d2ydx2−6(1+2x)dydx+16y=8(1+2x)2y(0)=0,y′(0)=2
[10M]
1996
1) Find the curves for which the sum of the reciprocals of the radius vector and polar subtangent is constant.
[8M]
8.(b) Solve: x2(y−px)=yp2,p=dydx
[15M]
2) ysin2xdx−(1+y2+cos2x)dy=0
[15M]
3) d2ydx2+2dydx+10y+37sin3x=0 Find the value of y when x=π/2, if it is given that y=3 and dydx=0, when x=0
[15M]
4) Solve: d4ydx4+2d3ydx3−3d2ydx2=x2+3e2x+4sinx
[15M]
5) Solve: x3d3ydx3+3x2d2ydx2+xdydx+y=x+logx
[15M]
1995
1) Determine a family of curves for which the ratio of the y-intercept of the tangent to the radius vector is a constant.
[20M]
2) Solve (2x2+3y2−7)xdx−(3x2+2y2−8)ydy=0
[20M]
3) Test whether the equation (x+y)2dx−(y2−2xy−x2)dy=0 is exact and hence solve it.
[20M]
4) Solve x3d3ydx3+2x2d2ydx2+2y=10(x+1x)
[20M]
5) Determine all real valued solutions of the equation y′′′⋅iy′′+y′⋅iy=0,y′=dydx
[20M]
6) Find the solution of the equation y′′+4y=8cos2x give that y=0 and y′=2 when x=0
[20M]