2008

1) Show that the functions y1(x)=x2 and y2(x)=x2logex are linearly independent. Obtain the differential equation that has y1(x) and y2(x) as the independent solutions.

[10M]

2) Solve the following ordinary differential equatjon of the second degree:
y(dydx)2+(2x−3)dydx−y=0

[10M]

3) Reduce the equation (xdydx−y)(x−ydydx)=2dydx to Clairaut’s form and obtain thereby the singular integral of the above equation.

[10M]

4) Solve:
(1+x)2d2ydx2+(1+x)dydx+y=4cosloge(1+x)

[10M]

5) Find the general solution of the equation
d2ydx2−cotxdydx−(1−cotx)y=exsinx

[10M]

6) Use the method of variation of parameters to solve the differential equation
x2d2ydx2+xdydx−y=x2ex.

[10M]

2007

1) Find the orthogonal trajectories of the family of the curves x2a7+y2b2+λ=1, λ being a parameter.

[10M]

2) Show that e2x and e3x are linearly independent solutions of

Find the general solution when y(0)=0 and dydx=1 at x=0.

[10M]

3) Find the family of curves whose tangents form an π4 angle with the byperbola xy=C.

[10M]

4) Apply. the method of variation of parameters to solve

[10M]

5) Solve

By using the method of removal of first derivate.

[10M]

6) Find the general solution of
(1−x2)d2ydx2−2xdydx+3y=0, if y=x is a solution of it

[10M]

2006

1) From x2+y2+2ax+2by+c=0, derive differential equation not containing a,b or c.

[10M]

2) Discuss the solution of the differential equation y2[1+(dydx)2]=a2.

[10M]

3) Solve
xd2ydx2+(1−x)dydx−y=ex

[10M]

4) Solve
d4ydx4−y=xsinx

[10M]

5) Solve
x2d2ydx2+xdydx−y=x2ex

[10M]

6) Reduce
xy(dydx)2−(x2+y2+1)dydx+xy=0 to Clairaut’s form and find its singular solution.

[10M]

2005

1) Form the differential equation that represents all parabolas each of which has latus rectum 4a and whose axes are parallel to the x−axis.

[10M]

2.(i) The auxiliary polynomial of a certain homogeneous linear differential equation with constant coefficients in factored form is P(m)=m4(m−2)6(m2−6m+25)3 What is the order of the differential equation and write a general solution?

[5M]

2.(ii) Find the equation of the one parameter family of parabolas given by y2=2cx+c2, c real and show that this family is self-orthogonal.

[5M]

3) Solve and examine for singular solution the following equation.

p2(x2−a2)−2pxy+y2−b2=0.

[10M]

4) Solve the differential equation d2ydx2+9y=sec3x.

[10M]

5) Given y=x+1x is one solution, solve the differential equation x2d2ydx2+xdydx−y=0 by reduction of order method.

[10M]

6) Find the general solution of the differential equation d2ydx2−2ydydx−3y=2ex−10sinx by the method of undetermined coefficients.

[10M]