# 
# Several Second Order differential equations are solved and checked.
# An auxiliary procedure LZ is used to facilitate the solutions.
# 
# Art Belmonte
# Mon, 27/May/96
# Math 308-509 [Maple V Release 4]
# Lecture, introducing:
# LZ: Set up for 2nd order homogeneous constant coefficient differential
# equations
# 
# Define the procedure LZ
> unprotect(LZ):
> LZ:=proc(c)
>   local a,d,k,L,n,p,r,s,x,y,Z; 
> n:=nops(c) - 1;
> a:=array(0..n, c);
> p:=0; s:=0;
> for k from 0 to n do 
>   p:=p + a[k]*r^(n-k);
>   s:=s + a[k]*(D@@(n-k))(y)
> od;
> L:=unapply(s, y); 
> Z:=unapply(p, r); 
> [eval(L), eval(Z)] 
> end: 
> protect(LZ):
# Ex 1 (S3. 1, T-120/7 )
> 
> unassign('y'); setup:=LZ([1, -9, 9]);
> L:=setup[1]; Z:=setup[2];
> deq:={convert(L(y)(x)=0, diff)};
> 

                      (2)                           2
     setup := [y -> (D   )(y) - 9 D(y) + 9 y, r -> y  - 9 y + 9]


                              (2)
                  L := y -> (D   )(y) - 9 D(y) + 9 y


                                   2
                        Z := r -> r  - 9 r + 9


                    / 2      \
                    |d       |     /d      \
            deq := {|--- y(x)| - 9 |-- y(x)| + 9 y(x) = 0}
                    |  2     |     \dx     /
                    \dx      /

> rt:=solve(Z(r)=0, r);

                                  1/2             1/2
                 rt := 9/2 + 3/2 5   , 9/2 - 3/2 5

> y:=unapply(c1*exp(rt[1]*x) + c2*exp(rt[2]*x), x);

                               1/2                          1/2
  y := x -> c1 exp((9/2 + 3/2 5   ) x) + c2 exp((9/2 - 3/2 5   ) x)

> check:=simplify(deq);

                           check := {0 = 0}

# Ex 2 (S3.1, T-120/11)
> unassign('y'); setup:=LZ([6, -5, 1]);

                       (2)                           2
    setup := [y -> 6 (D   )(y) - 5 D(y) + y, r -> 6 y  - 5 y + 1]

> L:=setup[1]; Z:=setup[2];

                                (2)
                  L := y -> 6 (D   )(y) - 5 D(y) + y


                                    2
                       Z := r -> 6 r  - 5 r + 1

> deq:={convert(L(y)(x)=0, diff)};

                      / 2      \
                      |d       |     /d      \
            deq := {6 |--- y(x)| - 5 |-- y(x)| + y(x) = 0}
                      |  2     |     \dx     /
                      \dx      /

> IC:={y(0)=4, D(y)(0)=0}; IVP:=deq union IC;

                    IC := {y(0) = 4, D(y)(0) = 0}


          / 2      \
          |d       |     /d      \
IVP := {6 |--- y(x)| - 5 |-- y(x)| + y(x) = 0, y(0) = 4, D(y)(0) = 0}
          |  2     |     \dx     /
          \dx      /

> rt:=solve(Z(r)=0, r);

                            rt := 1/3, 1/2

> y:=unapply(c1*exp(rt[1]*x) + c2*exp(rt[2]*x), x);

               y := x -> c1 exp(1/3 x) + c2 exp(1/2 x)

> eqs:=IC; sol:=solve(eqs, {c1, c2});

              eqs := {c1 + c2 = 4, 1/3 c1 + 1/2 c2 = 0}


                      sol := {c1 = 12, c2 = -8}

> y:=unapply(subs(sol, y(x)), x);

                y := x -> 12 exp(1/3 x) - 8 exp(1/2 x)

> check:=simplify(IVP);

                       check := {0 = 0, 4 = 4}

# Ex 3 (S3.4, T-142/11)
> unassign('y'); setup:=LZ([1, 6, 13]);

                     (2)                            2
    setup := [y -> (D   )(y) + 6 D(y) + 13 y, r -> y  + 6 y + 13]

> L:=setup[1]; Z:=setup[2];

                             (2)
                 L := y -> (D   )(y) + 6 D(y) + 13 y


                                  2
                       Z := r -> r  + 6 r + 13

> deq:={convert(L(y)(x)=0, diff)};

                   / 2      \
                   |d       |     /d      \
           deq := {|--- y(x)| + 6 |-- y(x)| + 13 y(x) = 0}
                   |  2     |     \dx     /
                   \dx      /

> rt:=solve(Z(r)=0, r);

                       rt := -3 + 2 I, -3 - 2 I

> y:=unapply(c1*exp(Re(rt[1])*x)*cos(Im(rt[1])*x)
>      + c2*exp(Re(rt[1])*x)*sin(Im(rt[1])*x), x);

       y := x -> c1 exp(-3 x) cos(2 x) + c2 exp(-3 x) sin(2 x)

> check:=simplify(deq);

                           check := {0 = 0}

# Ex 4 (S3.4, T-142/18)
> unassign('y'); setup:=LZ([1, 4, 5]);

                      (2)                           2
     setup := [y -> (D   )(y) + 4 D(y) + 5 y, r -> y  + 4 y + 5]

> L:=setup[1]; Z:=setup[2];

                              (2)
                  L := y -> (D   )(y) + 4 D(y) + 5 y


                                   2
                        Z := r -> r  + 4 r + 5

> deq:={convert(L(y)(x)=0, diff)};

                    / 2      \
                    |d       |     /d      \
            deq := {|--- y(x)| + 4 |-- y(x)| + 5 y(x) = 0}
                    |  2     |     \dx     /
                    \dx      /

> IC:={y(0)=1, D(y)(0)=0}; IVP:=deq union IC;

                    IC := {y(0) = 1, D(y)(0) = 0}


                  / 2      \
                  |d       |     /d      \
IVP := {y(0) = 1, |--- y(x)| + 4 |-- y(x)| + 5 y(x) = 0, D(y)(0) = 0}
                  |  2     |     \dx     /
                  \dx      /

> rt:=solve(Z(r)=0, r);

                         rt := -2 + I, -2 - I

> y:=unapply(c1*exp(Re(rt[1])*x)*cos(Im(rt[1])*x)
>      + c2*exp(Re(rt[1])*x)*sin(Im(rt[1])*x), x);

         y := x -> c1 exp(-2 x) cos(x) + c2 exp(-2 x) sin(x)

> eqs:=IC; sol:=solve(eqs, {c1, c2});

                   eqs := {-2 c1 + c2 = 0, c1 = 1}


                       sol := {c2 = 2, c1 = 1}

> y:=unapply(subs(sol, y(x)), x);

           y := x -> exp(-2 x) cos(x) + 2 exp(-2 x) sin(x)

> check:=simplify(IVP);

                       check := {1 = 1, 0 = 0}

# Ex 5 (S 3.5, 150/11)
> unassign('y'); setup:=LZ([9, -12, 4]);

                     (2)                              2
  setup := [y -> 9 (D   )(y) - 12 D(y) + 4 y, r -> 9 y  - 12 y + 4]

> L:=setup[1]; Z:=setup[2];

                              (2)
                L := y -> 9 (D   )(y) - 12 D(y) + 4 y


                                   2
                      Z := r -> 9 r  - 12 r + 4

> deq:={convert(L(y)(x)=0, diff)};

                    / 2      \
                    |d       |      /d      \
          deq := {9 |--- y(x)| - 12 |-- y(x)| + 4 y(x) = 0}
                    |  2     |      \dx     /
                    \dx      /

> IC:={y(0)=2, D(y)(0)=-1}; IVP:=deq union IC;

                    IC := {y(0) = 2, D(y)(0) = -1}


IVP :=

       / 2      \
       |d       |      /d      \
    {9 |--- y(x)| - 12 |-- y(x)| + 4 y(x) = 0, y(0) = 2, D(y)(0) = -1
       |  2     |      \dx     /
       \dx      /

    }

> rt:=solve(Z(r)=0, r);

                            rt := 2/3, 2/3

> y:=unapply(c1*exp(rt[1]*x) + c2*x*exp(rt[2]*x), x);

              y := x -> c1 exp(2/3 x) + c2 x exp(2/3 x)

> eqs:=IC; sol:=solve(eqs, {c1, c2});

                  eqs := {c1 = 2, 2/3 c1 + c2 = -1}


                      sol := {c1 = 2, c2 = -7/3}

> y:=unapply(subs(sol, y(x)), x);

              y := x -> 2 exp(2/3 x) - 7/3 x exp(2/3 x)

> check:=simplify(IVP);

                   check := {2 = 2, 0 = 0, -1 = -1}

# 
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