Lopez C.P. - Integral Calculus and Differential Equations using Mathematica

You can expand, factor and simplify polynomials andrational and trigonometric expressions, you can find algebraic solutions ofpolynomial equations and systems of equations, can evaluate derivatives andintegrals symbolically and find functions solution of differential equations,you can manipulate powers, limits and many other facets of algebraicmathematics series. Here are some examples ofsymbolic computation with Mathematica. 1) We can raise the bucketthe following algebraic expression: (x + 1) (x+2) (x+2) ^ 2. This is done bytyping the following expression: Expand [((x + 1) (x+2) (x+2) ^ 2) ^ 3]. Theresult will be another algebraic expression: In[1]: = Expand [((x + 1) *(x + 2) (x + 2) ^ 2) ^ 3]2 3Out[1] = 8- 12 x- 6 x- x 2) We can factor the resultof the calculation on the previous example by typing Factor [%] In[2]: = Factor [%]Out[2] = (2 + x) 3) We can resolve theindefinite integral of the function (x ^ 2) Sin(x) ^ 2 by typing Integrate [x ^2 Sin [x] ^ 2 x] In[3]:= Integrate[x^2*Sin[x]^2, x]Out[3]=3 2x x Cos[2 x] (1- 2 x) Sin [2 x]--- -------------+ -----------------------6 4 8 4) We can find the derivativeof the result of the integral above by typing D [% x] In[4]:= D[%, x] Out[4]= 2 2x Cos[2 x] (1 - 2 x ) Cos[2 x]--- -------------+-----------------------2 4 4 5) We can simplify theresult of the derivative before typing Simplify [%] In[5]:= Simplify[%]2 2Out[5]= x Sin[x] 6) We can develop in powerof order 14 series the result from the previous example by typing Series [% {x,0.14}] In[6]:= Series[%, {x, 0, 14}]Out[6]=6 8 10 12 144 x 2 x x 2 x 2 x 15x - -- + ---- - -----+-------- - ----------+ O[x]3 45 315 14175 467775 7) We can solve the equation 3ax - 7 x^ 2 + x ^ 3 = 0 (a, is a parameter) by typing Solve [3ax 7 x ^ 2 + x ^ 3 = 0] In[7]:= Solve[3*ax - 7*x^2 + x^3 == 0, x]Out[7]=1/37 49 2{{x->-- + ------------------------------------------------------------------+3 2 1/33 (686 - 81 ax + 9 Sqrt[-1372 ax + 81 ax ] )2 1/3(686 - 81 ax + 9 Sqrt[-1372 ax + 81 ax ] )+ ---------------------------------------------------------------},1/33 21/37 I -49 2{x-> - + - Sqrt[3] (----------------------------------------------------------+3 2 2 1/33(686-81ax+9Sqrt[-1372ax+81ax ] )2 1/3(686 - 81 ax + 9 Sqrt[-1372 ax + 81 ax ] )+ -----------------------------------------------------------------)-1/33 21/349 2- (---------------------------------------------------------------------+2 1/33 (686 - 81 ax + 9 Sqrt[-1372 ax + 81 ax ] )2 1/3(686 - 81 ax + 9 Sqrt[-1372 ax + 81 ax ] )+ ------------------------------------------------------------------)/ 2},1/33 21/37 I -49 2{x-> - - - Sqrt[3] (--------------------------------------------------------+3 2 2 1/33 (686-81ax+9 Sqrt[-1372ax+81ax ] )2 1/3(686 - 81 ax + 9 Sqrt[-1372 ax + 81 ax ] )+ ------------------------------------------------------------------)-1/33 21/349 2- (-----------------------------------------------------------------------+2 1/33 (686 - 81 ax + 9 Sqrt[-1372 ax + 81 ax ] )2 1/3(686 81 ax + Sqrt 9 [1372 ax + 81 ax])+ -----------------------------------------------------------------)/ 2}}1/33 2 8) We can find five complex solutionsof the equation x ^ 5 + 2 x + 1 = 0 by typing NSolve [x ^ 5 + 2 x + 1 = 0, x] In[8]:= NSolve[x^5 + 2*x + 1 == 0, x]Out[8]= {{x -> -0.7018735688558619 - 0.879697197929824 I},{x -> -0.7018735688558619 + 0.879697197929824 I},{x -> -0.486389035934543},{x -> 0.945068086823133 - 0.854517514439046 I},{{x > 0.945068086823133 + 0.854517514439046 I}} 9) We can generate a matrix3 x 3 whose (i, j) element is 1 / (i+j+1) by typing m = Table [1 / (i + j + 1),{i, 3}, {j, 3}] In[9]:= m = Table[1/(i + j + 1), {i, 3}, {j, 3}]Out[9] = {{1/3, 1/4, 1/5}, {1/4, 1/5, 1/6}, {1/5, 1/6, 1/7}} 10) We can invert thematrix above, typing Inverse [m] In[10]: = Inverse [m]Out[10] = {{300, 900, 630}, {900, 2880, 2100}, {630, 2100, 1575}} 11) We can find thedeterminant of the matrix m xIdentidad (3.3) tecleando Det[ m - xIdentityMatrix[3] ] In[11]:= Det[m - x*IdentityMatrix[3]]2 3Out[11] = (1 4755 x + 255600 x - 378000 x ) / 378000 12) We can find all thepermutations of the three elements {e, f, g} tecleando Permutations[{e, f, g}]