DE

Code: 2K1M5:9

PAGE 25

DIFFERENTIAL EQUATIONS, LAPLACE TRANSFORMS AND

FOURIER SERIES semester-vCode: 2K6M5:14RLevel : K

Unit : 1.1 Type : MCQ

1.The particular integral of is

(a) (b)

(c)

(d)

Code: 2K6M5:14RLevel : U


1.The complementary function of is

(a)

(b)

(c)

(d)

2.The C. F. of

(a) Ax6 + Bx-5

(b) Ax5 + Bx-1 (c) Ax-5 + Bx4

(d) Ax5 + Bx-53.The C. F. of

(a)

(b)

(c)

(d)

4.The C. F. of (x2D2 xD + 1) y = is

(a) (b) (A + B log x)x(c)

(d)

5.The C. F. of (x2D2 xD - 3) y = x2 is

(a) Ax(3 + Bx -(3 (b) Ax(3 + Bx(2(c) Ax(4 + Bx(2(d) Ax(3 + Bx-(46.The C. F. of (x2D2 + 8xD + 12) y = x4 is

(a) Ax-4 + Bx7

(b) Ax3 + Bx-4 (c) Ax2 + Bx4

(d) Ax2 + Bx57.The C.F. of (x2D2 + xD + 2) y = x2 is

(a) x(A coslog x + B sinlog x) (b) (A coslog x + B sinlog x)

(b) e-x (A coslog x + B sinlog x) (d) ey (A cos( log x) + B sin(log x))8.The P. I. of (x2D2 + xD + 2) y = x2 is

(a)

(b)

(c)

(d)

9.The P. I. of (x2D2 + 2xD + 2) y = 6x2 is

(a)

(b) x2

(c) x3

(d)

10.The P. I. of (x2D2 +8xD +12) y = x4 is

(a)

(b)

(c)

(d)

11. The P. I. of (x2D2 + xD - 3) y = x2 is

(a) x3

(b) x2

(c) x4

(d) x

Code: 2K6M5:14RLevel : K

Unit : 1.1 Type : VSA

1. If , what is the value of ?2.If & f()(0, what is the value of ?


Unit : 1.1 Type : PA

1. Solve (x2D2+3xD+1)y =

2.Solve (x2D2+4xD+2)y=ex.

3.Solve (x2D2+xD+3)y=x2.

4.Solve (x2D2+xD+2)y=x2.

5. Solve (x2D2+7xD+13)y=logx.

6. Solve (x2D2+2xD)y=6x2+2x+1.


Unit : 1.1 Type : E

1.(a) Solve (x2D2 + 3xD + 1) y =

(b) Solve (x2D2 + 4xD + 2) y = ex2.(a) Solve (x2D2 xD + 1) y =

(b) Solve (x2D2 + xD - 3) y = x23.(a) Solve (x2D2 + 8xD + 12) y = x4

(b) Solve (x2D2 + xD + 2) y = x24.(a) Solve (x2D2 + 2xD) y = 6x2 + 2x + 1

(b) Solve (x2D2 + 3xD + 4) y = x25.(a) Solve (x2D2 + 7xD + 13) y = log x

(b) Solve (x4D4 + 6x3D3 + 9x2D2 + 3xD +1) y = (1 + logx) 2Code: 2K6M5:14RLevel : U


1.The Auxiliary equation of (5 + 2x)2 is

(a) m2- 4m + 2=0(b) m2-2m+4=0(c) m2+4m+2=0(d) m2+2m+4=0

2.The A. E. of (3x+2)2

(a) m2-4=0

(b) m2+4=0

(c) m2+2=0

(d) m2-2=0

3.The A. E. of (1+x)2 2

(a) m2+1=0

(b) m2+2 = 0

(c) m2-1=0

(d) m2-2=0



1.Solve (5+2x)2

2.Solve (1+2x)2

3.Solve (x+a)2


Unit : 1.2 Type : E

1.Solve (3x+2)2

2.Solve (1+x)2

3.(x+a)2



1.The C. F. of (D2+n2) y = sec nx is

(a) A cos nx B sin nx

(b) A cos nx + B sin nx

(c) A sin nx + B cos nx

(d) A sin nx B cos nx

2.The C. F. of (D2 1) y = is

(a) Aex + Be2x

(b) Aex + Be-x

(c) Aex + Be3x3.The C. F. of y2 + y = cosec x is

(a) A cos x +Bx sin x

(b) A cos x + B sin x

(c) A cos x + B cos 2x

(d) B cos 2x A sin 2x

4. The C. F. of y2 + 4y = 4 tan 2x is

(a) A cos 3x + B sin 3x

(b) A cos 2x + B sin 2x

(c) A cos x + B sin x

(d) A cos 5x + B sin 5x



1.Find the C. F. of y2 + y = cosec x

2.Find the C. F. of y2 y =

3.Find the C. F. of y2 + n2y = sec nx

4.Find the C. F. of y2 + 4y = 4 tan 2x



1. Solve y2 + n2y = sec nx

2.Solve y2 + 4y = 4 tan 2x


Unit : 1.3 Type : E

1.y2 y =

2.y2 + y = cosec x

Code: 2K6M5:14RLevel :U

Unit : 2.1 Type : MC

1.The P.D.E. obtained by eliminating a from z = 2ax is

(a) 2p=z

(b) px=z

(c) ax=z

(d) 2x=z2.The P.D.E. obtained by eliminating a from z = 2ax+a2 is

(a) z = xp+p2 /4(b) z = 2px+p2 (c) z = xp+x2p2(d) xz = p+p2 /4

3.The P.D.E. obtained by eliminating a & b from z = (x+a)(y+b) is

(a) z = pq

(b) px = qy

(c) zp = xq

(d) zq = yx

4.The P.D.E. obtained by eliminating a & b from z = f(x2+y2) is

(a) px = pq

(b) pq = xy

(c) py = qx

(d) z = pq

5.The P.D.E. obtained by eliminating a & b from z = f(x3+y3) is

(a) py2 = qx2

(b) py = qx

(c) z = pq

(d) px2 = qy26.The P.D.E. obtained by eliminating a & b from z = ax + by + ab is

(a) z = px+qy+pq

(b) z = qx + py

(c) xz + xp + yq + pq = 0

(d) xz + yp = pq

7.The P.D.E. obtained by eliminating a & b from z = ax by + ab is

(a) z = px + qy - pq

(b) z = px qy + pq

(c) z = px + qy + pq

(d) pq = z

8.The P.D.E. obtained by eliminating a & b from z = ax + by + a is

(a) z = px + qy + p

(b) z = px + qy + q

(c) px = qy

(d) zq = xq + p+q

Code: 2K6M5:14RLevel :K


1.Define the order of partial Differential Equation.

2.By eliminating a, b from (x, y, z, a, b) = 0 What is the form of Partial Differential Equation?



1.What is the order of P.D.E. (z /(x + (z / (y = 0?

2.What is the order of P.D.E. (2z / (x2 + ((z / (y)2 +2z = 6

3.What is the order of P.D.E. (2z / (x2 + ((z / (x) +5z = 6

4.Obtain P.D.E. by eliminating a from z = 2ax.

5.Obtain P.D.E. by eliminating a from z = 5ax

6.Obtain P.D.E. by eliminating a & b from z = (x+a)(y+b)

7.Obtain P.D.E. by eliminating a & b from z = (x+2a) (y+2b)



10.Obtain P.D.E. by eliminating a & b from z = (x-2a) (y-3b)

11.Obtain P.D.E. by eliminating function f from z = f(2x-y)

12.Obtain P.D.E. by eliminating function f from z = f(3x+2y)

13.Obtain P.D.E. by eliminating function f from z = f(5x+6y).



1.Obtain a P.D.E. eliminating a, b from (x2 + y2) / a2 + (z2 / b2 )=1

2.Obtain a P.D.E. eliminating a, b from z = ax + by + ( (a2 + b2)

3.Obtain a P.D.E. eliminating a, b from (x-a)2 + (y-b)2 + z2 = a2 + b24.Obtain a P.D.E. eliminating a, b from z = (x2 + a) (y2 + b)

5.Obtain a P.D.E. eliminating a, b from a (x2 + y) + bz2 =1.

6.Obtain a P.D.E. eliminating a, b from z = ax + y ( (x2 ( a2) + bz2

7.Obtain a P.D.E. eliminating a, b from (x-a)2 + (y-b)2 + z2 = 1

8.Obtain a P.D.E. eliminating arbitrary function f from z = ey f (x + y)

9.Obtain a P.D.E. eliminating arbitrary function f from z = (x + y) f (x2 ( y2)

10.Obtain a P.D.E. eliminating arbitrary function f from z = ax + by +cz = f (x2 + y2 + z2)

11.Obtain a P.D.E. eliminating arbitrary function f from z = f (x2 + y2, z xy) =0

12.Obtain a P.D.E eliminating arbitrary function f from z = f (x2+y2+z2, x+y+z)=0.

13.Obtain a P.D.E eliminating arbitrary function f from z = f(z2 xy, x/z) =0

14.Obtain a P.D.E eliminating arbitrary function f from xyz = f(x + y + z)

15.Obtain a P.D.E eliminating arbitrary function f from z = y2 + 2 f(1 / x + log y)

16.Obtain a P.D.E eliminating arbitrary function f from f = (xy + z2, x + y + z) =0

17.Obtain a P.D.E eliminating arbitrary function f from

i) z = f(my lx)

ii) z = f(x2 y2)

18.Obtain a P.D.E eliminating arbitrary function f & g from z = f(y+ax)+xg(y+ax).

19.Obtain a P.D.E eliminating arbitrary function f & g from z = f(y) + g(x + y + z)

20.Obtain a P.D.E eliminating arbitrary function f & g from z = x f(y) + y g(x)

21.Obtain a P.D.E eliminating arbitrary function f & g from z = f(x+ay) + g(x-ay)

22.Obtain a P.D.E of all spheres whose centre lie on the plane z = 0 and whose radius is constant r

23.Obtain a P.D.E of all spheres whose centres lie on z axis

24.Obtain a P.D.E of all planes through the origin

25.Obtain a P.D.E of all planes having equal x & y intercepts

26.Obtain a P.D.E of all planes which are at a constant distance a from the origins


Unit : 2.1 Type : E

1.a) Obtain a P.D.E. eliminating a, b from (x2 + y2) / a2 + (z2 / b2 )=1

b) Obtain a P.D.E by eliminating arbitrary function f from z = ey f (x + y)

2.a) Obtain a P.D.E. eliminating a, b from z = ax + by + ( (a2 + b2 )

b) Obtain a P.D.E. eliminating arbitrary function f from z = (x + y) f (x2 ( y2)

3.a) Obtain a P.D.E. eliminating a, b from (x-a)2 + (y-b)2 + z2 = 1

b) Obtain a P.D.E. eliminating arbitrary function f from z=ax +by +cz =f(x2 +y2 + z2 ).

4.a) Obtain a P.D.E. eliminating a, b from a (x2 + y2) + bz2 =1.

b) Obtain a P.D.E. eliminating arbitrary function f from z = f (x2 + y2, zxy) =0

5.a) Obtain a P.D.E. eliminating a, b from x/a + y/a + z/b = 1 b) Obtain a P.D.E eliminating arbitrary function f from z = f(z2 xy, x/z) =0

6.a) Obtain a P.D.E of all spheres whose centre lie on the plane z = 0 and whose radius is constant r

b) Obtain a P.D.E eliminating arbitrary function f & g from z = f(x+ay)+ g(x-ay)

7.a) Obtain a P.D.E of all planes which are at a constant distance a from the origin.

b) Obtain a P.D.E eliminating arbitrary function f from f = (xy + z2, x+y+z) =0



1.The solution of ( z / ( x = ex is

a) z = ex + f(y)b) z = xex + f(y)c) z = ex + f(x)b) z = ex + xf(y)

2.The solution of x ( x/ ( y = 1 is

a) z = x2 + f(y) b) z = log x + f(y)c) z = log x + f(x)d) z=xlogx+ f(y)

3.The solution of y ( z / ( y = 5 is

a) z = 5log x + f(y)b) z = 5log y + f(x)c) z = 5/2 y2 + f(y)d) z=5logy + f(x)

4.The solution of x2 (z / (x = 6 is

a) z = -6/x + f(x)b) z = -6/x + f(y)c) z = 3x + f(y) d) z = 6/x + f(y)

5.The solution to be assumed while solving the equation p2 + q2 =4 is

a) p = a

b) q = a

c) q = ap

d) z =ax + by + c



1.Define singular solution

2.Define complete solution

3.Define general solution

4.What is the solution to be assumed while solving PDE f(p, q) = 0

5.How do we assume a solution for PDE f1(x, p) = f2 (y, q)

6.The solution to be assumed while solving the equation f(p, q) = 0 is

a) p = a

b) q = ac) z = ax + by + cd) p = q = a

7. The solution to be assumed while solving the equation f1(x, p) = f2(y, q) is

a) f1(x, p) = f2(y, q) = ab) p = a c) q = a

d) q = ap



1.Write the solution of ( z / ( x = sin x

2.Write the solution of ( z / ( x = ex3.Write the order of x ( z / ( x = 1

4.Write the order of y ( z / ( y = 5

5.Write the order of x2 ( z / ( x = a

6.Write the order of y ( z / ( y = 6

7.Write the equation q p + x y = 0 in the form f1(x, p) = f2(y, q)

8.Write the equation p + q = sin x + sin y in the form f1(x, p) = f2(y, q)

9.Write the equation p2 y(1 + x2) = qx210.Write the equation p + x = qy in the form f1(x, p) = f2(y, q)

11.Write p + q = px + qy in the form f1(x, p) = f2(y, q)



1.Solve p2 + q2 = 4

2.Solve p + q = x + y

3.Solve x ( z / ( x = 2x + y 4.Solve p + q = sinx +siny

5.Solve (2 z / ( x2 = xy + 3x

6.Solve p + x = qy

7.Solve x ( z / ( x = 2x + y

8.Solve (2 z / ( x ( y = x2 + y2 9. Solve p = q2

10. Solve ( p + ( q = 2x and (2 z / ( x ( y = cosx* cosy

11. Solve p2 2q2 = 4pq

12. Solve pq + p + q = 0

13. Solve p + q = 2x and x + y ( z / ( x = 0

14. Solve xp y2 q2 = 1 and (2 z / ( y2 = siny

15. Solve pq = xy


Unit : 2.2 Type : E

1.Solve i) pq = 1

ii) pq + p + q = 0

2.Solve i) pq = 4

ii) pq + p + 2q = 0

3.Solve i) q2 3p + q = 0

ii) p3 + q3 = 0

4.Solve i) 3p2 ( 2 q2 = 4pq

ii) p2 + q2 = npq

5.Solve i) p + q = x + y

ii) ( p + ( q = 2x

6.Solve i) p2 + q2 = x + y

ii) ( p + ( q = x

7.Solve i) p + q = sin x + sin y

ii) pq = xy

8.Solve i) 2yp2 = q

ii) xp y2 q2 = 1

9.Solve i) p2 y (1 + x2) = qx2

ii) py + qx = pq

10.Solve pq + qx = y

ii) p + q = px + qy

11.Solve i) p + x = qy

ii) py + qx = pq



1.The tentative solution for f(x, p, q) = 0 is

a) x = constantb) y = constantc) p = constantd) q = constant

2.The tentative solution for f(y, p, q) = 0 is

a) x = constantb) y = constantc) p = constantd) q = constant

3.The tentative solution for f(z, p, q) = 0 is

a) x = ap

b) y = aq

c) p = aq

d) q = ap

4.The Clairauts form of P.D.E is

a) z = xp + yq + f(x, y)

b) z = xp + qy + f(p, q)

c) z = xp yq f(p, q)

d) z = xp + yq f(p, q)

5.Lagranges equation is obtained by eliminating

a) u & v from f(u, v) = 0

b) p & q from f(p, q) = 0

c) x & y from f(x, y) = 0

d) x & z from f(x, z) = 0

6.u(x, y, z) = c1 & v(x, y, z) = c2 are the solutions of the equations

a) dx / R = dy / P = dz / Q

b) dx / Q = dy / P = dz / R

c) dx / P = dy / Q = dz / R

d) dx / p = dy / q = dz / r



1.For solving f(z, p, q) = 0, the substitution used is:

a) p=a

b) q=a

c) q=apd) p=q

2.For solving 9(2pz+3q)=z2, the substitution used is:

a) p=a

b) q=a

c) p=q+ad) p=aq

3.For solving z=pq, the substitution used is:

a) p=a

b) q=a+pc) q=apd) p=q24. The tentative solution for z = pq + 3q is

a) p =ab) q = ac) q = apd) p = aq



1.Write Lagranges equation

2.Write Clairauts form of P.D.E ?

3.What is the solution for Clairauts form of P.D.E ?



1.Explain the procedure of solving the equation f (x, p, q) = 0 and hence solve p+q=x2.Explain the procedure of solving the equation f (y, p, q) = 0 and hence solve q=2yp23.Explain the procedure of solving the equation f (z, p, q) = 0 and hence solve z=p2+q2Code: 2K6M5:14RLevel :U


1.Solve completely i) q = xp + q2

ii) p = y2 q22.Solve completely i) p = 2qx

ii) q = 2yp23.Solve completely 9(p2 z + q2) = 4

4.Solve completely i) p(1 + q) = qz

ii) pq = x

5.Solve completely 4(1 + z3) = 9z4 pq

6.Solve completely i) p2 z2 + q2 = 1

ii) q2 = yp47.Solve completely i) (p + (q = x

ii) z = p2 + q28.Solve completely i) p2 z2 + q2 = 1

ii) (p + (q = y

9.Solve completely i) p + q = z

ii) z2 (p2 + q2 + 1) = a210.Solve completely i) p(1+q2) = q(z-1)

ii) p2 = qz

11.Solve completely z = px + qy + pq

12.Solve completely z = px + qy + p/q-p

13.Solve completely (1-x) p + (2-y) q = 3-z

14.Solve completely z/pq = x/q + y/p + ( pq

15.Solve completely (y2 + z2) p- xyq = -xz

16.Solve completely (y + z) p + (z + x) q = x + y

17. Solve completely x2 p + y2 q = (x + y) z

18.Solve completely (x2 yz)p + (y2 zx)q = z2 - xy

19.Solve completely xp + yq = z

20.Solve completely xp - yq = xy

21.p tan x + q tan y = tan z

22.(y-z)p + (z-x)q = x-y

23.Solve x(y-z)p + y(z-x)q = z(x-y)

24.Solve x(y2 z2) + y(z2 x2) q = z(x2 - y2)

25.Solve xzp + yzq = xy

26.Solve up xq + x2 y2 = 0

27.Solve z(xp yq) = y2 x2

28.Solve (mz ny)p + (nx lz)q = ly - mx

29. Solve p q = log ( x + y)

30.Solve (xz + yz)p + (xz yz)q = x2 + y2

31.(x2 y2 z2)p + 2xyz = 2xz

32.(yz/x) p + xzq = y233.Solve (y + z x)p + (z + x y)q = x + y - z

34.Solve y2 zp = x2 zq = x2 y

35.Solve z = px + qy + 2 ( pq.


Unit : 2.3 Type : E

1.a) Explain the procedure of solving the equation f(x, p, q) = 0

b) Derive Lagranges equation.


Unit : 2.3 Type : E

1.a) Solve i) p = 2qx

ii) 9(p2 z + q2) = 4

b) Solve (y + z)p + (z + x)q = x + y

2.a) i) p = y2 q2

ii) p(1+q) = qz

b) Solve x(y2- z2) + y(z2 x2)q = z(x2 y2) = 0

3.a) q = 2yp2

ii) z = p2 + q2

b) Solve x(y z)p + y(z x)q = z(x y)

4.a) Solve i) q = xp + p

ii) pz + q = 1

b) Solve (mz ny)p + (nx lz)q = ly mx

5.a) Solve z = px + qy + p/q - p

b) Solve xzp + yzq = xy

6.a) Solve z = px + qy = 2(pq

b) p tan x + q tan y = tan z

7.a) Solve z px + qy + pq

b) Solve yp xq + x2 y2 = 0

8.a) Solve z/pq = x/q + y/p + (pq

b) Solve (xz + yz)p + (xz yz)q = x + y

9.a) Solve z = px + qy + 3 (pq

b) Solve y2 z p x2 z q = x2 y



1.L(eat) is

a) b)

c)

d)

2.Le-at is

a)

b)

c) d)

3.L (coshat) is

a) b) c)

d)

4.L (sinhat) is

a)

b)

c)

d)

5.L (cosat) is

a)

b)

c)

d)



1.Define Laplace transform



1.Find L(t3 + 2t + 3)

2.Find L(t3 3t2 + 2)

3.Find L(at2 + bt + c)

4.Find L(t2 5t 7)



1.Prove that L(f`(t)) = s2 L(f(t)) sf(0) f1 (0)



1.Find L{f(t)}, where f(t) = 0 when 0 1

= 0 when t < 4

3.Find the Lapalce transform of

f (t) = e-1 when 0 4

4.Find the Laplace transform of

f(t) = sin t when 0

DE

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