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MA2261 MATHEMATICS II
UNIT I – Ordinary Differential EquationHigher order linear differential equations with constant coefficients – Method of variation of parameters – Cauchy’s and Legendre’s linear equations – Simultaneous first order linear equations with constantcoefficients.
1.Solve the following differentaial equations :
(a ) d2 y
dx 2− 3
dydx
− 4 y = 0
(b )d3 ydx3
− 3d2 ydx2
+ 3dydx
− y = 0
(c ) d 4 ydx 4
− 5d2 ydx2
+ 4 y = 0
(d )d4 y
dx4+ 8
d2 y
dx2+ 16 y = 0
(e )d3 ydx3
+ 6d2 ydx2
+ 11dydx
+6 y = 0
( f )d2 y
dx2+ 4
dydx
+29 y = 0 , given when x = 0 , y = 0 anddydx
= 1 . 5
2.Solve the following differential equations :
(a )d3 y
dx3+ y = 3+5ex
(b )d2 y
dx2− 4 y = ( 1+ex )2
(c )d2 ydx 2
+ 4dydx
+5 y = −2 cosh x
3. Solve the following differential equations :(a ) (D−2 )2 y = 8 (e2 x+sin 2 x )(b )( D2−4 D+3 ) y = sin 3 x cos 2 x
(c )( D3+1) y = sin 3x − cos 2( x2 )
4 . Solve the following differential equations :(a ) (D−2 )2 y = 8x2
(b )d2 y
dx2+
dydx
= x2+2 x+4
(c )d2 ydx 2
+ y = e2 x+ cosh 2 x + x3
Dr. D. Saravanan, Professor of Mathematics
2
5. Solve the following differential equations :
(a ) d2 y
dx2− 4 y = x sinh x
(b ) d 4 ydx 4
− y = cos x cosh x
(c ) (D2−2 D ) y = ex sin x(d ) ( D2 + 4 D +8) y = 12 e−2 x sin x sin 3 x
(e ) d2 y
dx 2+ 2 y = x2e3 x+ ex cos 2 x
6 . Solve the following differential equations :
(a )d2 y
dx2− 2
dydx
+ y = x ex sin x
(b ) ( D2 − 4 D + 4 ) y = 8 x2 e2 x sin 2 x
(c )d2 y
dx 2+ 4 y = x sin x
7 . Solve the following differential equations by var iation of parameters :
(a )d2 y
dx2+ 4 y = 4 sec2 (2 x )
(b ) d2 ydx 2
+ y = cosec x
(c ) d2 y
dx2+ y = sec x
(d ) d2 ydx2
+ 4 y = tan 2 x
(e ) d2 ydx 2
+ y = x sin x
8 . solve the following Cauchy ' s hom ogeneous linear differential equations :
(a ) x3d3 ydx 3
+ 2 x2d2 ydx2
+ 2 y = 10 (x+1x )
(b ) x2d2 y
dx2− x
dydx
− 3 y = x2 log x
(c ) x2d2 ydx 2
+ xdydx
+ y = log x sin ( log x )
(d ) x2 d2 ydx2
− 3 xdydx
+ 5 y = sin ( log x )
(e ) x2 d2 y
dx2+ 2 x
dydx
− 12 y = x3 log x
Dr. D. Saravanan, Professor of Mathematics
3
9 . Solve the following Legendre ' s linear differential equations :
(a ) (3 x+2 )2d2 y
dx 2+ 3 (3 x+2 ) dy
dx− 36 y = 3 x2+4 x+1
(b ) (2 x+1 )2d2 ydx2
− 6 (2 x+1 )dydx
+ 16 y = 8(1+2 x )2
(c ) (2 x+3 )2d2 ydx2
− 2 (2 x+3 ) dydx
− 12 y = 6 x
10. Solve the following simul tan eous differential equations :
(a )dxdt
+ 4 x + 3 y = 1 ;dydt
+ 2 x + 5 y = e t
(b )dxdt
= 7 x − y ;dydt
= 2 x + 5 y
(c ) dxdt
+ y = sin t ;dydt
+ x = cos t ; given x=2 , y=0 when t=0
UNIT II - VECTOR CALCULUS
Gradient Divergence and Curl – Directional derivative – Irrotational and solenoidal vector fields – Vector integration – Green’s theorem in a plane, Gauss divergence theorem and stokes’ theorem (excluding proofs) – Simple applications involving cubes and rectangular parallelpipeds.
Grad, Curl, Divergence
1. Find the unit normal vector at the point (2, -2, 3) to the surface x2y + 2xz = 42. Find the unit normal vector to the surface x2y + 2xz2 = 8 at (1, 0, 2)3. Find the directional derivative of = x2yz + 4xz2 + xyz at (1, 2, 3) in the
direction 2i + j – k4. Find the maximum directional derivative of = 3x2 + 2y - 3z at (1, 4, 1) in the
direction 2i + 2j – k5. Find the angle between the surfaces x2 + y2 + z2 = 9 and z = x2 + y2 - 3 at
(2, 1, -2).6. Find the angle between the surfaces x2 + y2 = 4 – 5z and x2 + y2 + 3z2 = 104
at (5, 2, -5).7. Find the angle between the normals at (1, 1, 1) and (4, 1, 2) to the surface xy
– z2 = 0.8. Find the angle between the normals at (3, 3, -3) and (4, 1, 2) to the surface
xy = z2.9. Find the angle between the tangent planes to the surfaces x log z = y2 - 1, x2
y = 2 – z at (1, 1, 1).10. For what value of ‘ k ‘ is the vector rk r solenoidal.
Dr. D. Saravanan, Professor of Mathematics
4
11. Find ‘m’ , if (2 x+ y ) i + (4 x−11 y+3 z ) j + (3 x+mz )k is solenoidal.
12. Determine f(r) so that f (r ) r is solenoidal
13. Show that F = (2xy+z3 ) i + x2 j + 3 xz { k ¿ is conservative. Find such that = F.
14. Show that F = ( y2 cos x+z3 ) i + (2 y sin x−4 ) j + 3 xz2 k is irrotational. Also find the scalar potential.
15. Prove that ∇( 1
r2) =−2 R
r 4
16. Prove that ∇( 1
rr ) =−2 r
r3
17. Prove that ∇(r n) = nr n−2 r18. Prove that div (rnR) = (n+3)rn
19. Prove that curl (rnR) = 020. Prove that div(curl F ) = 021. Prove that curl (grad ) = 022. Prove that curl curl F = grad div F - 2F
23. If F = x2 y i − 2 xz { j + 2 yz { k ¿¿ , find curl curl F.
24. Show that ∇2 ( f (r )) = f ' ' (r ) + 2
rf ' (r )
25. ∇ .(ϕ ∇ ψ − ψ ∇ ϕ )= ϕ ∇2ψ − ψ ∇ 2 ϕ
Line Integral
26. Evaluate
∫ F . d r from (0 , 0 , 0 ) to (1 , 1 , 1) along x = t , y = t2 , z = t3 and F=(3 x2+6 y ) i − 14 yz { j + 20 xz 2 k ¿
27. Evaluate the line integral ∫c
( x2+xy )dx + ( x2+ y2 )dy where C is the square
formed by the lines y = 1 and x = 1.
28. Evaluate the work done by the force F = (5 xy−6 x2 ) i +( 2 y−4 x ) j along the curve C in the xy-plane, y = x3 from (1, 1) to (2, 8).
29. F = 5 xy { i + 2 y j ¿ Evaluate ∫c
F . d r where C is the part of the curve y = x3 between
x = 1 and x = 2
30. Prove that ∫c
F . d r is independent of path where
F = ( y2 cos x + z3 ) i +( 2 y sin x−4 ) j + 3 xz2 k Also evaluate the integral from (0, 0, 0) to (/2, 0. 1)
Dr. D. Saravanan, Professor of Mathematics
5
Surface Integral
31. Evaluate ∫( F . n )d s
where F = 6 z i − 4 j + y k and S is the portion of the plane 2x + 3y +6z =12 in the first octant.
32. F = y i + 2 j + xz { k m /sec .¿ Show that the flux of water through the parabolic cylinder y = x2, 0 x 3, 0 z 2 is 69 m3/sec.
33. Evaluate ∬
s
Curl F . n ds where F = ( x2− y2) i + 2 xy { j ¿ over the surface of the
rectangle in the plane z = 0 and bdd by the line x = 0, y = 0, x = a, y = b.
34. F = y i + z j + xz { k m /sec .¿ Show that the flux of water through the parabolic cylinder y = x2, 0 x 3, 0 z 2 is 75 m3/sec.
Green’s Theorem
35. Evaluate ∫c
( xydx −x2dy by converting this into a double integral. It is given
that C is the bounding of the region bdd byy = x and x2 = y.
36. Prove that area bdd by simple closed curve is given by
12∫( xdy − ydx ) .
Hence find the area bounded by the parabola y2 = 4ax and its latus rectum.
37. Find the area bounded by the ellipse
x2
a2+ y2
b2= 1 .
38. Using Green’s Theorem in plane, evaluate ∫c
(( y − sin x )dx + cos x dy ) where C
is the triangle OAB whose vertices are O(0, 0), A((/2, 0), B(/2, 1)
39. Apply Green’s Theorem in the plane to evaluate ∫c
(3 x2 − 8 y2 )dx + ( 4 y − 6 xy ) dy
where C is the boundary of the region defined by x = 0, y = 0 and x + y = 1
Stokes Theorem
40. Verify Stokes Theorem for F = ( y−z+2) i +( yz+4 ) j − xz { k ¿ where S is the surface of the cube x = 0, x = 2, y = 0, y = 2, z = 0, z = 2 above the XOY plane.
41. Verify Stokes Theorem for the region z = 0 plane bounded by x = 0, x = a, y =
0 and y = b for F = ( x2− y2 ) i + 2xy { j . ¿
Dr. D. Saravanan, Professor of Mathematics
6
42. Verify Stokes Theorem for A =(2 x− y ) i − yz2 j − y2 z k . where S is the upper half of the sphere x2 + y2 + z2 = 1 and C is its boundary.
43. Verify Stokes Theorem for A = y i +2 yz { j+ y2 k . ¿ taken over the upper half of surface S of the sphere x2 + y2 + z2 = 1, z 0 and the bounding circle x 2 + y 2 = b, z = 0 .
44. A = y i + z j +x k upper half of sphere x2 + y2 + z2 = 1 and the boundary C of the circle x2 + y2 + z2 = 1, z = 0
Gauss – Divergence Theorem
45. Verify GDT for F = x2 i + y2 j +z2 k taken over the cube bounded by x = 0, x = 1, y = 0, y = 1, z = 0, z = 1.
46. Verify GDT for F = ( x2− yz ) i +( y2−zx ) j +(z2−xy ) k taken over the surface S bounded by x = 0, x = a, y = 0, y = b, z = 0, z = c.
47. Using GDT evaluate ∫s
F . n ds where F = 4 xz { i − y2 j + yz { k ¿ ¿ and S is the surface
of the cube bounded by x = 0, x = 1, y = 0, y = 1, z = 0, z = 1.
48. Evaluate ∫s
F . d s where F = 4 x i −2 y2 j+z2 k and S is the surface bounded by
the region x2 + y2 = 4, z = 0 and z = 3.
49. If S is any closed surface enclosing Volume V and F = x i +2 y j +3 z k then
show that ∬ F . n ds= 60
UNIT III - ANALYTIC FUNCTIONS
Functions of a complex variable – Analytic functions – Necessary conditions, Cauchy – Riemann equation and Sufficient conditions (excluding proofs) – Harmonic and orthogonal properties of analytic function – Harmonic conjugate – Construction of analytic functions – Conformal mapping : w= z+c, cz, 1/z, and bilinear transformation.
1. Find the analytic region of f(z)=(x-y)2 +2i (x+y).
2. Examine whether ez is analytic.
3. Examine whether |z|2 is analytic.
4. Examine whether z2 & z3 are analytic.
5. Properties of analytic function.
Dr. D. Saravanan, Professor of Mathematics
7
6. If f(z) is analytic prove that ( ∂2
∂ x2+ ∂2
∂ y2 )|f ( z )|2=4|f ' ( z )|2 .
7. If u=ex [ xcos y− y sin y ] ,find f (z).
8. If u( x , y )=3 x2 y+2 x2− y3−2 y2,find v, f (z).
9. Determine f (z) if v=sin 2x
cosh 2 y−cos2 x.
10. Find f (z) if v=3 x2 y− y3
11. Find f(z) if u=log √x2+ y2
12. Verify v=( xcos y− y sin y ) ex is harmonic. Construct analytic function.
13. If u−v=ex (cos y−sin y ) . find f (z).
14. If u−v= (x− y ) ( x2+4 xy+ y2 ) find f(z).
15. If 2 u+v=e2 x [ (2 x+ y ) cos2 y+( x−2 y )sin 2 y ] .Find f (z).
16. If 2 u+v=ex (cos y−sin y ) ., find f (z).
17. If f (z) is regular prove that ( ∂2
∂ x2+ ∂2
∂ y2 )|Re f ( z )|2=2|f '( z )|2 .
18. If u & v are harmonic prove that ( ∂u∂ y
−∂ u∂ x )+i(∂ u
∂ x+ ∂ u∂ y )
is analytic.
19. Give an example that both u & v are harmonic but f (z) is not analytic.
20. If f(z) is analytic in the complex plane prove that ( ∂2
∂ x2+ ∂2
∂ y2 ) log|f ( z )|=0 .
21. Find the analytic function whose imaginary part is ex2−2 y2
sin(2 xy ).
22. If v=log ( x2+ y2) is harmonic. Find the Real part of analytic function with this
function v as its Imaginary part.
23. Prove that function with constant modulus is constant.
24. Prove that analytic function with constant real part is constant.
Dr. D. Saravanan, Professor of Mathematics
8
25. Stream functionψ=tan−1 ( y
x ). Find velocity potential.
26. If velocity potential =3x2y-y3. Find Stream function.
CONFORMAL MAPPING
27. Find the image of the y-axis under the transformationw=z2.
28. Discuss the transformationw=1
z .
29.Draw image of square (0,0) (1,0) (1,1) &(0,1) under w =(1+i)z.
30. Find the image of |z−3 i|=3 under the mappingw=1
z .
31. Image of |z+2 i|=2 underw=1
z .
32. Find the image if |z|=2 under w=z+3+2i
33. Find the image of |z−2 i|=2 ifw=1
z .
34. Image of x2− y2=1 , is ρ2=cos2 φ under
w=1z .
35. Image of x=k underw=1
z .
FIXED POINTS, CRITICAL POINTS
36. Find the critical points of w=z2
37. Find the fixed points ofw= z+4
z+1 .
38. Find the fixed points of w= z−1−i
z+2
39. Find the critical points of w=z4−4 z
40. Find the critical points ofw=( z−α ) ( z−β )
Dr. D. Saravanan, Professor of Mathematics
9
BILINEAR TRANSFORMATION
41. Find the bilinear transformation which maps the points (-i, 0, i) onto the points (-1, i,
1) respectively
42. Find the bilinear transformation which maps the points z=1,i,-1 onto i,0,-i
43. Find the bilinear transformation which maps the points (-1,i,1) onto (1,i,-1).
44. Find the bilinear transformation which maps the points (1,i,-1) onto (0,1,).
45. Show that this transformation maps interior of unit circle to upper half w plane.
46. Find the bilinear transformation which maps the points (-1, 0, 1) to (0, i, 3i).
47. Given (-1,0,1) maps to (-1,-i,1). Show that this maps the upper half of the z-plane
onto interior of unit circle |w|=1
48. Discuss the invariant points of this transformation, (0, 1,) to (-5,-1,-3)
UNIT IV - COMPLEX INTEGRATION
Complex integration – Statement and applications of Cauchy’s integral theorem and Cauchy’s integral formula – Taylor and Laurent expansions – Singular points – Residues – Residue theorem – Application of residue theorem to evaluate real integrals – Unit circle and semi-circular contour(excluding poles on boundaries).
Evaluate the following Using Cauchy integral formula
1. where C : |z|=1
2. where C |z| =3
3. where C |z| =3
4. where |z| = 1
Dr. D. Saravanan, Professor of Mathematics
∫c
e2 z
z−3dz
∫c
cos πz2
( z−2 )( z−5 )dz
∫c
cos πz2+sin πz2
(z−1)2( z−2)dz
∫c
z2+1( z−2 )( z−3 )
dz
1( z+1) z
10
5. around |z-i| = 2
6. where |z| =1
7. where |z| = 2
8. whereC is |z| = 1
9. where Cis |z| =1
10. C : |z-1|=3
11. Find f(z) & f’(z) Where C: |z| = 2.5
Taylor’s & Laurent Series
12. about i) z= 0 ii) z=1
13. in |z| =1
14. in |z+1| < 1
15. in |z| <1 & |z| >4
16. in |z|>2 & 0<|z-1| <1
17. in |z|<2, 2< |z|< 3, |z| > 3
Dr. D. Saravanan, Professor of Mathematics
∫c
ez
z−πdz
z2−4( z+1)( z+4 )
∫c
ez dz
∫c
cos πzz−1
dz
∫c
dzz−1
∫c
z+1z (2 z+1 )
dz
∫c
ez
( z+1)2dz
f (a )=∫c
2 z2−z−2( z+a)
dz
z−1z+1
1( z+1)( z+2)
z2−1( z2+5 z+6 )
1( z−1 )(z−2)
11
18. in 3 < |z+2| < 5
19. in 1 < | z+1 | < 3
20. in the nbd of singular point
singular points and zeros
21.sin (1/z)
22.sin(1/(1-z))
23. (tan z)/z
24.cos (/z)
25.ez
26. .
27.
Residue Theorem
Z=a is a simple pole z lim a (z-a) f(z)
If z=a is a pole of order m
Coefficient of (z-a)-1 in the expansion of f(z) around
an isolated singularity
28. f(z) =
z
( z−1 )2( z+2) at the singular point z=1
29. where C is |z| = 4
Dr. D. Saravanan, Professor of Mathematics
z2−6 z−1( z−1 )(z−3)( z+2)
ez
( z−1 )2
z3−1z3+1
cos πz
( z−a )2
7 z−2( z+1) z (z−2)
z lim adm−1
dzm−1( z−a )m f ( z )
∫c
1
( z2+a)2dz
12
30.∫c
z+7
z2+2 z+5dz
where C: |z-i| = 1.5
31.∫c
zsec z
1−z2dz
where C: 4x2 + 9y 2 = 36
32.∫c
zsec z
(1−z )2dz
C: |z| = 3
33.∫c
cos πz2+sin πz2
(z−1)2( z−2)dz
where C: |z| =3
34.∫c
z+4
z2+2 z+5dz
where C : |z +1-i| = 2
35.∫ e2 z
( z+1)4dz
where C : |z| =2
36.∫c
12 z−7
( z−1 )2(2 z+3)dz
where C: |z| = 2
37.∫ dz
( z2+4 )2where C : | z –i| = 2
38.∫ zdz
( z−1 )( z−2)2 C : |z-2| = ½
39.∫c
ez
z2+1 dz C: |z| = 2
Dr. D. Saravanan, Professor of Mathematics
13
UNIT V - LAPLACE TRANSFORMATION
Laplace transform – Conditions for existence – Transform of elementary functions – Basic properties – Transform of derivatives and integrals – Transform of unit step function and impulse functions – Transform of periodic functions. Definition of Inverse Laplace transform as contour integral – Convolution theorem (excluding proof) – Initial and Final value theorems – Solution of linear ODE of second order with constant coefficients using Laplace transformation techniques.
Part A
1. L(t n) exists if n > ------- s> -------------
2. L(K) is ----------
3. Define periodic function and state its Laplace transform formula.
4. Find L (sin (2t + 3 ).
5. True or false: L{(f(t)} = (s) then L{f( t / 2 )} = 2 (2s).
6. If f(t) is a periodic function of period 2, then L{f ( t)} =--------------------
7. L{ t 3/2 + cos t + 1}
8. State the first shifting theorem on Laplace transform
9. Give two examples of functions which have no L.T
10. The period of | sin t | is………….
11. L(f(t) = 1 / (s-2) 2 then tlt
0 f(t) = ----
12. State the condition for which the L.T of f(x) exist
13. If L (f(t)) = (s) then L (f ”(t))------
14. L({et – e-3t}/t)
15. If L{(f(t)} = F(s) then show that L{f(at)} = 1/a F(s/a)
16. L {∫0
∞
te−3 t sin 2 tdt}
17. L ( t/e t)
Dr. D. Saravanan, Professor of Mathematics
14
18. L{S an e-bt cos nt }
19. If f(t) = e -2t sin 2t find L (f ’(t))
20. Write the value of 1 * e t
21. Write the value of t * e t
22. If f(t) = t ½ find L(f(t))
23. L {0∫t e –t dt }
24. If L (f(t)) = 1/ s(s+2) then verify the initial & final value theorems
25. Give an example of a function which has L.T but it is not continuous
26. Another name of the unit step function is ----------
27. L(f(t) = 1 / (s-2) 2 then tlt
0 f(t) = ----
28. L-1( 2/ s 2)
29. If L-1((s))= f ( r ) then L -1
30. L-1 ( 1/ (s + a ) is valid for…………..
31. L-1{cot -1 ( 2/(s+1)}
32. L -1(1) = ------------
Part B
Find Laplace transform of the following
1. t e-2t sin3t
2.
cos2 t−cos3 tt
3. cos 32t
4.
sin wt 0<t< πw
0πw
<t<2 πw
5. e –t ∫
sin tt
6. Prove that L[f"(t) ]=s2Lf(t)-sf (0) – f ' (0).
7. sinh 3t cos2 t
8. t2 e -3t sin 2t
9. t sin t
10. (sin at) / t
11.
Dr. D. Saravanan, Professor of Mathematics
φ( s )s
f ( t )={ t 0<t<a2 a −t a< t <2 a
and f ( t +2a )=f ( t )
15
12. L (f(t)=(s) then L( f(t) / t 2 ) =-----
13. Write the value of t * e t
14. (e at - cos bt) / t
15.
16. (e -3t sin 2t )/t
17. sinh 2 2t
18. e –t tn
19. A full sine wave rectifier given by f(t) = E sin wt in 0 < t < /w and f(t + /w) = f(t)
20. cos 3 2t
21. (cos 2t – cos 3t) / t
22.f ( t )={ t 0<t <2
4−t 2<t <4and f ( t+4 )=f ( t )
23.
24. (e-t sin t)/t
25.
[ t∫0
t
e−4 t cos3 tdt ]+sin 5 tt
26.
Find Inverse
27. L -1 {(s + 1)/ .(s2 + s + 1) }
28. L -1 {(e-3s)/(s-2)4 }
29. L -1 {s/((s2 + 1)(s2 + 4))}
30. L -1 {1/ ( s 2 + 4s + 4 )}
31. L-1 {(5s + 3)/(s 2 + 2s+ 5)}
32. L -1 {(2s + 3)/ (2s2 + 6s + 13)2}
33. L-1 {(1/ s4 – 1)}
Dr. D. Saravanan, Professor of Mathematics
f ( t )={ sin t 0< t<π2 π −t π< t <2 π
andf ( t +2π )=f ( t )
f ( t )= { t 0< t<π2 π −t π<t <2 π
and f ( t+2 π )=f ( t ) is1
s2tanh
πs2
f ( t )={ E 0≤t≤T /2−E T /2< t<T
and f ( t+T )=f ( t )
16
34. L-1 {(e –s/ s2)}
35. L-1 {(3s+2)/(3s2 + 4s + 3)2}
36. If L ( ∫ f(t))=(s) then L f(t)dt =(s)/s Use this result to find L-1{1/s(s+a)}
37. L-1 ((s)) = f(t) , find L -1 (e-as (s))
38. L-1 {cot -1 ((s+3)/2)}
39. L -1 { s/(s+2)3}
40. L-1 { s / (s 2 +1)( s 2 +4)}
41. L-1{ e -2s/ s(s+1)}
42. L-1{s2 / (s2 + a2)2}
43. L-1 {(1/ s4 + 4a4)}
Using Convolution theorem
44. State and Prove the convolution theorem
45. L -1 s/(s2+1)2
46. L -1 1/(s2+4)2
47. L -1 1/ ( s2 + a2)2
48. L -1 s2 / (s2 +a2)( s2+b2)
49. L-1{ 1/ ((s+1)(s+9)2)}
50. L-1 { 1/(s+1)(s+2)}
51. L-1 { 1/(s3(s+5)}
52. L-1{ 1/ s2(s2 +25)}
53. L -1 s / (s2 +9)( s2+125)
Application of L.T.- Solve the following differential equations
54. y”+4y=sin2t, given y (0)= y’(0)= 0.
55. y” – 2 y’ + 2y = 0 y = y’ = 1 at x = 0
56. y” -2y’ + x = e –t x( 0) = 2 x’(0) =1
57. y”–y’-2y = 20 sin 2t given y(0) = 0 y’(0) = 2
58. y” + 9 y = 18 t given y(0) = 0 = y(/2)
59. y” + 4y = f(t) with y(0) = 0 and y’(0) = 1 and f(t) = t if 0 < x < 1
and 0 else where.
60. y” + y’ = t2 + 2t given y = 4 and y = -2 at t = 0
61. y” – 3y’ + 2y = e –t given y(0) = 1 & y’(0) = 0
Dr. D. Saravanan, Professor of Mathematics
17
62. y’’ + 2y’ -5y = e-t sin t given y(0) = 0 and y’(0) = 1
63. y” + y’ -2y = 3 cos 3t – 11 sin 3t given y(0) = 0 y’() = 6
64. y” + 2y’ + y = t e-t given y’(0) = -2
65. y” + 6y’ + 9y = 2 e-3t given y(0) = 1 & y’(0) = -2
66. y” + 4y’ +4y = t e-t given y(0) = 0 ,y’(0) = -1
67. y” + 3y’ + 2y = 2( t2 + t + 1) given y(0) = 2 y’(0) = 0
68.
69. Evaluate integral ∫ e-2t sin3t dt
70. Evaluate ∫ e-t cos 2t dt
71. Solve: y (t) = t 2 + y(x) ∫ sin ( t-x) dx
Dr. D. Saravanan, Professor of Mathematics
∫0
∞e−t−e−2t
tdt