My Mathematica Tutorial Differential Equations

Mathematica student guide Haris Javed

The Purpose of this Guide is to inform the fellow students how to use mathematica for a Linear Algebra and Differential Equation Coarse: H* H* H* H* H* H* H*

We Can Write Comments in mathematica by using H* Text*L command. Many of the Functions can Shift + enter should be used to Obtain a Function, just hitting enter will just take you to We can Assign Variables as a Function, For example typing n=2 will give n the value of 2, Also You Can Also Use In@xD or Out@xD To bring or use old commands, just put the number in x, for Pallettes Can Also be used instead of typing commands, you can use pallettes by clicking Pallettes Pi can be Written as  pi  without spaces or Π*L infinity can be Written as  inf  or ¥ *L

2

test mathematica.nb

1 Solving Solving an Equation With Mathematica

In[2]:=

SolveA4x-x2 Š 6,xE H*note that x2 Requires Ctrl+6, and there are two equal signs Š between the

Out[2]=

::x ® 2 - ä

In[21]:=

SolveA5x2 -18x-5Š0,xE

Out[21]=

In[22]:=

Out[22]=

::x ®

1 5

2 >, :x ® 2 + ä

J9 -

106 N>, :x ®

2 >>

1 5

J9 +

106 N>>

SolveA-6+3x-2x2 +4x3 Š 5 SinHxL,xE ::x ®

32 - 60 Sin

1 -

+ 13

6

12 223 299 + 45 Sin + 3

2

3

3387 + 570 Sin + 875 Sin - 500 Sin

3

13

299 + 45 Sin + 3

3

3387 + 570 Sin + 875 Sin2 - 500 Sin3

>,

6 213 :x ®

J1 + ä

1

3 N H32 - 60 SinL

+

13

6 24

J1 - ä

223

299 + 45 Sin + 3

3 N 299 + 45 Sin + 3

3

2

3

3387 + 570 Sin + 875 Sin - 500 Sin

13

3

3387 + 570 Sin + 875 Sin2 - 500 Sin3

>,

12 213 :x ®

J1 - ä

1

3 N H32 - 60 SinL

+ 6

13

24 223 299 + 45 Sin + 3

J1 + ä

3 N 299 + 45 Sin + 3

3

2

3

3387 + 570 Sin + 875 Sin - 500 Sin

13

3

3387 + 570 Sin + 875 Sin2 - 500 Sin3 12 213

>>

test mathematica.nb

In[23]:=

Out[23]=

H*by Adding A N to the Function in input@26D We can change all the exact Value to Approximate SolveA-6+3x-2x2 +4x3 Š 5 SinHxL,xEN 0.0524967 H32. - 60. SinL

::x ® 0.166667 -

+ 13

299. + 45. Sin + 5.19615

2

3

3387. + 570. Sin + 875. Sin - 500. Sin

13

0.132283 299. + 45. Sin + 5.19615

3387. + 570. Sin + 875. Sin2 - 500. Sin3

13

299. + 45. Sin + 5.19615

2

3

3387. + 570. Sin + 875. Sin - 500. Sin

H0.0661417 - 0.114561 äL 299. + 45. Sin + 5.19615

13

>,

13

>>

3387. + 570. Sin + 875. Sin2 - 500. Sin3

H0.0262484 - 0.0454635 äL H32. - 60. SinL

:x ® 0.166667 +

13

299. + 45. Sin + 5.19615

2

3

3387. + 570. Sin + 875. Sin - 500. Sin

H0.0661417 + 0.114561 äL 299. + 45. Sin + 5.19615

Out[24]=

>,

H0.0262484 + 0.0454635 äL H32. - 60. SinL

:x ® 0.166667 +

In[24]:=

3387. + 570. Sin + 875. Sin2 - 500. Sin3

H* we Can Also Add a . in the Function itself to Create decimalApproximate Values. Note the . SolveA-6.+3x-2x2 +4x3 Š 5 SinHxL,xE H0.0524967 - 0.090927 äL H32. - 60. SinL

::x ® 0.166667 +

13

2392. + 360. Sin + 929.516

2.92889 - 1. Sin

2

2.31282 + 1.17889 Sin + Sin

H0.0330709 + 0.0572804 äL 13

2392. + 360. Sin + 929.516

2.92889 - 1. Sin

2.31282 + 1.17889 Sin + Sin2

>,

H0.0524967 + 0.090927 äL H32. - 60. SinL

:x ® 0.166667 +

13

2392. + 360. Sin + 929.516

2.92889 - 1. Sin

2

2.31282 + 1.17889 Sin + Sin

H0.0330709 - 0.0572804 äL 13

2392. + 360. Sin + 929.516

2.92889 - 1. Sin

2.31282 + 1.17889 Sin + Sin2

>,

0.104993 H32. - 60. SinL

:x ® 0.166667 -

+ 13

2392. + 360. Sin + 929.516

2.92889 - 1. Sin

2

2.31282 + 1.17889 Sin + Sin

13

0.0661417 2392. + 360. Sin + 929.516 In[34]:=

Out[34]=

3

Solve@87x-4yŠ-8,-1x+3yŠ18
48

118 ,y®

17

17

>>

2.92889 - 1. Sin

2.31282 + 1.17889 Sin + Sin2

>>

4

test mathematica.nb

In[94]:=

Out[96]=

H*Multiple instructions can also be given for Solving a System*L Clear@equD; equ=89x+2yŠ-3,-2x+7yŠ11<; Solve@equD Solve@N@equDD NSolve@equD ::x ® -

43

93 ,y®

67

67

>>

Out[97]=

88x ® - 0.641791, y ® 1.38806<<

Out[98]=

88x ® - 0.641791, y ® 1.38806<<

In[102]:=

Out[102]=

Clear@equD;equ=82x-1y-3zŠ-4,-5x+2y-18zŠ2,4x+y+2zŠ-3<;Solve@equD Solve@N@equDD NSolve@equD ::x ® -

162

127 ,y®

145

43 ,z®

145

145

>>

Out[103]=

88x ® - 1.11724, y ® 0.875862, z ® 0.296552<<

Out[104]=

88x ® - 1.11724, y ® 0.875862, z ® 0.296552<<

In[117]:=

SolveAx2 +2x-7Š0,xE

Out[117]=

::x ® - 1 - 2

In[118]:=

N@Out@117DD

Out[118]= In[131]:=

2 >, :x ® - 1 + 2

2 >>

88x ® - 3.82843<, 8x ® 1.82843<< Clear@equD;equ=83 x-2 y-7 z==-4,-3 x+2 y-8 z==1,2 x+11 y+6 z==-3<;Solve@equD Solve@N@equDD NSolve@equD

test mathematica.nb

In[134]:=

ContourPlot3D@83 x-2 y-7 z==-4,-3 x+2 y-8 z==1,2 x+11 y+6 z==-3<,8x,-5,5<,8y,-5,5<,8z,-5,5
0

-5 5

Out[134]=

0

-5 -5

0

5

5

6

test mathematica.nb

2

Roots

Finding Roots Using the FindRoot Command In[25]:=

Out[25]=

H* mathematica is a Powerful tool which can be used for many operations, now lets see how we can FindRootA4x3 -5x2 +3x-3Š0,8x,0<E 8x ® 1.16058<

H* above is the Excat root, it does not include all the roots that are found by using the Solve PlotA94x3 -5x2 +3x-3=,8x,-10,10<,AxesLabel ® 8x,y<,PlotRange®8-10,10<EH* below we can see the Root In[27]:=

Out[27]= In[28]:=

H*Another Root*L FindRootA4x3 +2x2 +x-5Š3Sin@xD,8x,2<E 8x ® 1.03301<

H*Visual Roots of Above Equation*L PlotA94x3 +2x2 +x-5,3Sin@xD=,8x,-10,10<,AxesLabel ® 8x,y<,PlotRange®8-10,10<E y 10

5

Out[28]=

-10

-5

5

10

x

-5

-10 In[135]:= Out[135]=

FindRootA2x5 -5x4 -2x3 -x2 +6x+1Š3Sin@xD-4Cos@xD,8x,3<E 8x ® 2.81381<

*L

test mathematica.nb

3

7

Graphing

Plotting A Basic Equation In[3]:=

H*We will Graph the above equation Eq1 here*L H*The above equation will be Defined as 2 functions @y=6D and Ay= 4x-x2 E put together as follows y

5

Out[3]=

-4

-2

2

4

-5

Graphing A Little More complex function In[4]:=

Out[4]=

H* Lets Solve a Little More Complex Function*L SolveA5x2 -18x-5Š0,xE ::x ®

1 5

J9 -

106 N>, :x ®

1 5

J9 +

106 N>>

x

8

test mathematica.nb

::x ®

J9 + 106 N>> 5 5 H*Also Lets Plot This function and see how it looks like*L PlotA5x2 -18x-5,8x,-3,3<,AxesLabel ® 8x,y<,PlotRange® 8-8,8<E 1

J9 -

106 N>, :x ®

1

y

5

-3

-2

-1

1

2

3

x

-5

In[5]:=

H* We will Plot another function just to get the hang of the method*L PlotA92x4 +x3 -7x2 +3x-8=,8x,-20,20<, AxesLabel® 8x,y<,PlotRange ® 8-20,20<E

H* The image Below can be Shrinked, or Expanded with the use of the Mouse cursors. Click on the y 20

10

Out[5]=

-20

-10

10

-10

-20

20

x

test mathematica.nb

In[30]:=

9

PlotASin@xD2 ,8x,0,2Pi<,PlotLabel® Sin@xD2 E

sin2 HxL

1.0

0.8

0.6 Out[30]=

0.4

0.2

1

2

3

4

5

6

Plotting With Different Style H* the Original Function Was

2 3

6 2 1 1 3 x- ==1- x*L PlotB: x- ,1- x>,8x,-.5,1.5<,PlotRange® 8-.5,1.2< 3 2 3 3 2

1.0

0.5 Out[6]=

-0.5

0.5

1.0

1.5

-0.5

In[7]:=

PlotB:4-x,

H5-6xL 2

>,8x,-2,4
H5-6xL 2

*L

5

Out[7]=

-2

-1

1

2

3

4

-5

Plotting a three Dimensional function , Graphing a system of Equations with x,y,z or x1, x2, x3

10

test mathematica.nb

Plotting a three Dimensional function , Graphing a system of Equations with x,y,z or x1, x2, x3

In[8]:=

ContourPlot3D@84x-8y-zŠ-4,-6x+3y-5zŠ1,6x+y+6zŠ-12<,8x,-10,10<,8y,-10,10<,8z,-10,10
0 5

0 10 10

-5

-10 5

0 Out[8]=

-5

-10

test mathematica.nb

In[151]:=

Plot3DB:

H18x-2y+7L H-36+4y-1L H3+2x+yL , , >,8x,-5,3<,8y,-2,4<,PlotRange® 8-1,2<,PlotStyle® 8Red 4 1 -6

2

Out[151]=

11

1

4

0 2 -1 -4 0

-2 0 2

-2

12

test mathematica.nb

3-D Plotting H*Mathematica will plot 3d Graphs With ease look at the Following Example*L Plot3D@Sin@x+y^2D,8x,-3,3<,8y,-2,2<,Mesh® AllDH*This Plot can be moved Scaled by clicking on

1.0 0.5 Out[18]=

0.0

2

-0.5 -1.0

1 -2 0 0 -1

2 -2

In[19]:=

Plot3D@Sin@x+y^2D,8x,-2,2<,8y,-2,2<,PlotStyle®Directive@Orange,Specularity@White,20DDD H*with

*

test mathematica.nb

1.0 0.5 0.0 -0.5 -1.0 -2 2

-1 1

0 0

1

-1

2

-2

13

14

test mathematica.nb

In[20]:=

H*Note that contour plot will plot a 3d function if we have more than 2 variables, if we only ContourPlot@88x+2yŠ-3,-8x+3yŠ11<,8x,-6,3<,8y,-2,4<,Axes® True,Frame® False,Mesh® AllD 4

3

2

Out[20]=

1

-6

-4

-2

2

-1

-2

test mathematica.nb

In[138]:=

Out[138]=

GraphPlot3D@Table@i -> Mod@i^2, 75D, 8i, 75 None, MultiedgeStyle -> None, EdgeRenderingFunction -> H8Green, Cylinder@ð, 0.1D< &L, VertexRenderingFunction -> H8Black, Sphere@ð, 0.2D< &LD

15

16

test mathematica.nb

In[174]:=

p5 = ContourPlot By Š

H5xL 2 +

2x2

J N 1 2

, 8x, -1, 15<, 8y, 0, 1.5<, FrameTicks ® TrueF

1.4

1.2

1.0

0.8 Out[174]=

0.6

0.4

0.2

0.0 0

5

10

15

test mathematica.nb

4

17

Matrix

RowReduction, Determinant, Transpose, Inverse In[31]:=

H* For row reduction we will just use the Command Row Reduce*L 8 -5 -4 -4 RowReduceB -5 2 -11 1 F 7 1 3 -3

Out[31]=

In[32]:=

::1, 0, 0, -

41 87

>, :0, 1, 0, -

4 87

>, :0, 0, 1,

10 87

>>

2 -5 -7 -4 RowReduceB -3 2 -15 1 FMatrixForm 2 1 6 -3

Out[32]//MatrixForm=

1 0 0 0 1 0 0 0 1

352 163 79 163 49

163

NBRowReduceB Out[33]= In[36]:=

Out[36]=

4 3 2 FFH* again the N gives Decimal Answers*L 2 3 4

881., 0., - 1.<, 80., 1., 2.<<

H* We can always Define a Variable in mathematica as a function, look below *L 4 3 2 n= 2 3 4 884, 3, 2<, 82, 3, 4<<

H* We have defined n As the Matrix above, be careful about lower and uppercase. A lot of Commands H* We can Clear the above n by using the Clear@nD command*L Clear@nD

In[61]:=

H* A Equation can Also be put as a linear system and be solved, 89x+2yŠ-3,-2x+7yŠ11< can be put lets solve them and see if the answers match*L Solve@equD

Out[61]=

::x ® -

43

93 ,y®

67

67

>>

18

test mathematica.nb

In[62]:=

Out[62]=

RowReduceB

9 2 -3 F -2 7 11

::1, 0, -

>, :0, 1,

43 67

93 67

>>

H* The Answers Match, as expected *L In[105]:=

Clear@equD

In[109]:=

2 -5 -7 Clear@m,u,v,wD;m= -1 8 -5 ; 9 4 6 u=8-8,1,-2<;v=812,4,4<; w=83,5,-8<;f=LinearSolve@mD

Out[111]= In[112]:=

LinearSolveFunction@83, 3<, <>D H* Transposing the Above matrix*L Transpose@8f@uD,f@vD,f@wD
Out[112]//MatrixForm=

- 704 863

1148 863 - 100 863 1080 - 863

353 863 533 863

In[113]:=

- 434 863 122 863 - 581 863

8u,v,w<MatrixForm

Out[113]//MatrixForm=

-8 1 -2 12 4 4 3 5 -8 In[114]:=

Join@m,8u,v,w
Out[114]//MatrixForm=

2 -5 -1 8 9 4 -8 1 12 4 3 5 In[115]:=

-7 -5 6 -2 4 -8

n=Transpose@Append@Append@Append@Transpose@mD,uD,vD,wDDMatrixForm

Out[115]//MatrixForm=

2 - 5 - 7 - 8 12 3 -1 8 -5 1 4 5 9 4 6 -2 4 -8 In[116]:=

RowReduce@Out@115DDMatrixForm

Out[116]//MatrixForm= 704

1148

863 353

863 100

1 0 0 0 1 0 0 0 1

863 533 863

-

863 1080 863

-

434

863 122 863 581

-

863

test mathematica.nb

In[119]:=

Out[119]=

In[121]:=

Out[121]=

Clear@A,A1,b,cD;A=889,7,3<,88,5,1<,87,5,5<<;b=8-2,5,3<;c=81,-1,0<;A1=MatrixForm@889,7,3<,88,5 ::-

1

1 ,-

2

5

, 0>, :-

1

1 ,

2

1 ,

5

3

>, 82, 5, 3<, 8Log@2D, Log@5D, Log@3D<, 6>

H* Dot Product Cross Product *L 8b c,b*c,b´c,b.c,Dot@b,cD,Cross@b,cD,b×c,b‰c,A.b,b.A,MatrixForm@bD< :8- 2, - 5, 0<, 8- 2, - 5, 0<, 8- 2, - 5, 0<, - 7, - 7, 83, 3, - 3<, 8- 2, 5, 3< × 81, - 1, 0<, 83, 3, - 3<, 826, 12, 26<, 843, 26, 14<,

In[122]:=

94+A,3+A2,Transpose@AD,Transpose@A1D,AT ,A2 ,A12 =MatrixForm 8813, 11, 7<, 812, 9, 5<, 811, 9, 9<< 3 + A2 889, 8, 7<, 87, 5, 5<, 83, 1, 5<< 9 7 3 TransposeB 8 5 1 F 7 5 5

Out[122]//MatrixForm=

999T , 7T , 3T =, 98T , 5T , 1=, 97T , 5T , 5T ==

8881, 49, 9<, 864, 25, 1<, 849, 25, 25<< 9 7 3 8 5 1 7 5 5 In[123]:=

- 3 5 19 0 1 5 6 - 18 27

Out[124]= In[126]:=

H* Determinant*L Det@Out@123DD - 315 B+1MatrixForm

Out[126]//MatrixForm=

- 2 6 20 1 2 6 7 - 17 28 In[128]:=

2

Clear@BD;MatrixForm@B=88-3,5,19<,80,1,5<,86,-18,27<
Out[123]//MatrixForm=

In[124]:=

19

BT MatrixForm

Out[128]//MatrixForm=

H- 3LT 0T 6T

5T 1

19T 5T

H- 18LT 27T

-2 5 > 3

20

test mathematica.nb

In[130]:=

Transpose@BDMatrixForm

Out[130]//MatrixForm=

-3 0 6 5 1 - 18 19 5 27 In[216]:=

Inverse@BDMatrixForm

Out[216]//MatrixForm=

-

13

53

35 2

35 13

21 2

21 8

105

In[218]:= Out[218]= In[219]:= Out[219]= In[222]:=

105

-

2 105 1 21 1

105

LinearSolve@BD LinearSolveFunction@83, 3<, <>D NullSpace@BD 8< Eigenvalues@N@BDDMatrixForm

Out[222]//MatrixForm=

27.5269 - 4.87447 2.34761 In[226]:=

Eigenvectors@N@BDDSimplifyMatrixForm

Out[226]//MatrixForm=

0.54023 0.155871 0.826956 0.986349 0.10673 - 0.125397 - 0.877241 - 0.46351 - 0.124926

test mathematica.nb

In[147]:=

Out[147]=

Graphics3D@8 AstronomicalData@ð, "OrbitPath"D & ž AstronomicalData@"Planet"D, 8Orange, [email protected], HAstronomicalData@ð, "OrbitPath"D & ž AstronomicalData@"ApolloAsteroid"DL< <, PlotRange -> 88-6, 6<, 8-6, 6<, 8-4, 4<<, Boxed -> False, ViewAngle -> А11D

21

22

test mathematica.nb

5 derivatives and integrals Differentiation H*Remember to Always clear your Variables*L Clear@xD;Clear@nD In[161]:=

D@xn ,xD

Out[161]=

n x-1+n

In[162]:= Out[162]=

D@xn ,x,8x,3
In[163]:=

DB

,xF 3Tan@xD

Out[163]=

x2 Cot@xD -

1

x3 Csc@xD2

3 Manipulate@D@2xn ,xD,8n,1,20,1
n Out[167]=

2

Integration In[164]:=

IntegrateAx2 ,xE x3

Out[164]=

3 In[165]:=

2 à x âx

x3 Out[165]=

3

test mathematica.nb

23

Manipulate@Integrate@2xn ,xD,8n,1,20,1
n Out[166]=

2 x3 3

1 In[168]:=

IntegrateB

,xF 1-x3

ArcTanB

F

1+2 x 3

1 Log@- 1 + xD + 3

3

LogA1 + x + x2 E

1

-

Out[168]=

6 1

In[170]:=

Out[170]=

ManipulateBIntegrateB

,xF,8n,1,20,1
:1.63758 ´ 10-15 ,

n

2 J5 +

1 2 20

5 N ArcTanB

1-

2 J5 +

1+ 2

10 - 2

5

5 +4x

ArcTanB 10 - 2

J- 1 +

In[171]:=

à

5 N LogB1 -

1 2

J- 1 +

5 N

F+

F - 4 Log@- 1 + xD -

5 N x + x2 F + J1 +

âx x3 -1 1+2 x 3

3

F

1 +

1 Log@- 1 + xD -

3

6

>

5

1

ArcTanB Out[171]=

5 +4x

LogA1 + x + x2 E

5 N LogB1 +

1 2

J1 +

5 N x + x2 F

24

test mathematica.nb

In[172]:=

Plot@%,8x,0,2
-1.2

-1.4 Out[172]=

-1.6

0.5

1.0

1.5

2.0

test mathematica.nb

6

25

vector

fields and differential equations Plotting Vector field Graphs H* Note Mathematica 7 uses VectorPlot command instead of VectorFieldPlot In[183]:=

VectorPlotA9ã-12x Sin@2yD, 2y Cos @5xD=, 8x,-Π, Π<, 8y, -Π, Π<, Axes ® True, AxesLabel® 8x,y<, y

3

2

1

Out[183]=

-3

-2

-1

1

-1

-2

-3

2

3

x

26

test mathematica.nb

In[187]:=

VectorPlot@8Sin@6yD, -y Cos @6xD<, 8x,-Π, Π<, 8y, -Π, Π<, Axes ® True, AxesLabel® 8x,y<, Frame y

3

2

1

Out[187]=

-3

-2

-1

1

2

3

x

-1

-2

-3

In[192]:=

VectorPlot @8Sin@2xD,Cos@yD<,8x,-Π, Π<,8y,-Π,Π<,Axes® True,AxesLabel® 8x,y<, Frame® TrueD y

3

2

1

Out[192]=

0

x

-1

-2

-3

-3

-2

-1

0

1

2

3

H* Lets Solve some differential Equations now we will use DSolve Command which Solves 2 Simultaneous DSolve@8y@xDŠ -z'@xD,z@xDŠ -y'@xD<,8y,z<,xDH* This equation involves no boundary conditions*L Out[190]=

::z ® FunctionB8x<,

1 2

ã-x I1 + ã2 x M C@1D 1

y ® FunctionB8x<, 2

1 2

ã-x I- 1 + ã2 x M C@2DF,

ã-x I- 1 + ã2 x M C@1D +

1 2

ã-x I1 + ã2 x M C@2DF>>

D

test mathematica.nb

In[191]:= Out[191]= In[193]:= Out[193]= In[194]:= Out[194]=

DSolve@8y'@xDŠ a y@xD,y@0DŠ 1<,y@xD,xD 88y@xD ® ãa x << DSolve@2y@tD+2y'@tD+y''@tD==2t,y@tD,tD@@1DD 9y@tD ® - 1 + t + ã-t C@2D Cos@tD + ã-t C@1D Sin@tD= y@tD. DSolve@2y@tD+2y'@tD+y''@tD==2t,y@tD,tD@@1DD - 1 + t + ã-t C@2D Cos@tD + ã-t C@1D Sin@tD

In[199]:=

Clear@fD

In[203]:=

f@t_D:=t+

Out[204]= In[205]:= Out[205]=

27

c1 Cos@5tD c2 Sin@5tD + -9 ã-t ã-5t f@tD.8c1® 1,c2® 1<

- 9 + t + ãt Cos@5 tD + ã5 t Sin@5 tD Plot@Table@f@xD,8c2,-2,2<,8c1,-2,2
20

10

-3

-2

-1

1

2

3

> -10

-20

-30

-40

28

test mathematica.nb

In[206]:=

Plot@Evaluate@Table@f@xD,8c2,-2,2<,8c1,-2,2
100

Out[206]=

:0.765,

-2

-1

1

2

3

>

-100

-200

In[227]:= Out[227]=

NDSolve@8x'@tDŠ -3Hx@tD-y@tDL,y'@tDŠ -x@tDz@tD+26.5x@tD-y@tD,z'@tDŠ x@tDy@tD-z@tD,x@0DŠ z@0D 88x ® InterpolatingFunction@880., 200.<<, <>D, y ® InterpolatingFunction@880., 200.<<, <>D, z ® InterpolatingFunction@880., 200.<<, <>D<<

test mathematica.nb

In[228]:=

29

ParametricPlot3D@Evaluate@8x@tD,y@tD,z@tD<.%D,8t,0,200<,PlotPoints® 100000DTiming -10

-5

0

5

10

40

Out[228]=

:2.75,

>

30

20

10 10

0

0

-10

In[207]:=

y@tD.DSolve@2y@tD+2y'@tD+y''@tD==2Sin@tD,y@tD,tD@@1DDSimplify 4

Out[208]=

5

+ ã-t C@2D Cos@tD +

2 5

+ ã-t C@1D Sin@tD

30

test mathematica.nb

In[209]:= Out[209]=

In[211]:=

Out[211]=

y@tD.DSolve@2y@tD+2y'@tD+y''@tD==2Sin@tD,y@tD,tD@@1DDN 2.71828-1. t C@2D Cos@tD + 2.71828-1. t C@1D Sin@tD + 0.2 H- 5. Cos@tD + Cos@tD Cos@2. tD - 2. Cos@2. tD Sin@tD + 2. Cos@tD Sin@2. tD + Sin@tD Sin@2. tDL y@tD.DSolveAy''@tD-4y'@tD+3y@tDŠ ãH3tL -5Cos@3tD,y@tD,tE@@1DDSimplify ãt C@1D + ã3 t C@2D +

1 12

In[212]:=

Out[212]=

In[214]:=

Out[214]=

I3 ã3 t H- 1 + 2 tL + 2 Cos@3 tD + 4 Sin@3 tDM

H*Initial Conditions*L DSolveA9y''@xD-4y'@xD+3y@xDŠ ãH3xL -5Cos@3xD,y@0DŠ1,y'@0DŠ1 =,y@xD,xESimplify ::y@xD ®

1 6

I9 ãx - 4 ã3 x + 3 ã3 x x + Cos@3 xD + 2 Sin@3 xDM>>

DSolveAx2 y''@xD+y@xDŠ 0,y@xD,xE ::y@xD ®

1

1 3 Log@xDF +

x C@1D CosB 2

x C@2D SinB

3 Log@xDF>> 2