# The DE is S'=rS-k, with r=0.04 and k=(150)*(12), with S(0)=9000: Eqn1:=diff(S(t),t)=0.04*S(t)-(150)*12;

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AmtMoney:=rhs(dsolve( { Eqn1, S(0)=9000},S(t)));

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# Paid in full? Set the equation to zero- Since we're in Maple, we can also solve it. solve(AmtMoney=0,t);

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# The DE will be P'=kP with P(0)=50. The solution is easy enough to write directly, but in Maple: Eqn2A:=diff(P(t),t)=k*P(t);

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJQRzYiNiNJInRHRilGKyomSSJrR0YpIiIiRidGLg==

P1:=rhs(dsolve({Eqn2A,P(0)=50},P(t)));

LCQtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomSSJrR0YoIiIiSSJ0R0YoRiwiI10=

# Now solve for k- We know that t=2 gives 100 fish: Eqn2B:=subs(t=2,P1)=100;

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kk:=solve(Eqn2B,k);

LCQtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIjIyIiIkYq

# Final population of fish so far: P2:=subs(k=kk,P1);

LCQtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiYtSSNsbkdGJTYjIiIjIiIiSSJ0R0YoRjAjRjBGLyIjXQ==

# If we harvest removing k fish per year, then: Eqn2C:=diff(P(t),t)=kk*P(t)-k;

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJQRzYiNiNJInRHRilGKywmKiYtSSNsbkc2JEYlSShfc3lzbGliR0YpNiMiIiMiIiJGJ0Y0I0Y0RjNJImtHRikhIiI=

# Since we're using Maple, we can also go ahead and solve (this is linear and separable): dsolve({ Eqn2C,P(0)=50 },P(t));

Ly1JIlBHNiI2I0kidEdGJSwmKiYtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjIiIjISIiSSJrR0YlIiIiRjAqJilGMCwkRicjRjNGMEYzLCYiI11GM0YpISIjRjNGMw==

# Interesting that Maple chose to move into base 2 instead of base e! #Cannonball: mv'=mg-gam v, or v'=g-(gam/m) v m:=10; g:=9.8; gam:=5;

IiM1

JCIjKSohIiI=

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Eqn3:=diff(v(t),t)=g-(gam/m)*v(t);

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ2RzYiNiNJInRHRilGKywmJCIjKSohIiIiIiJGJyNGLyIiIw==

# Note that initial velocity is -10 (minus for "up") v1:=rhs(dsolve({Eqn3,v(0)=-10},v(t)));

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# Just for fun, we can plot the velocity to see if it makes sense (and it seems to!). plot(v1,t=0..10);

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# Tanks! Eqn4A:=diff(Q(t),t)=9-(3/100)*Q(t);

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJRRzYiNiNJInRHRilGKywmIiIqIiIiRicjISIkIiQrIg==

dsolve({Eqn4A,Q(0)=0},Q(t));

Ly1JIlFHNiI2I0kidEdGJSwmIiQrJCIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCRGJyMhIiQiJCsiISQrJA==

# This looks reasonable, since Q(0)=0 and as t-> infinity, Q -> 300. Eqn4B:=diff(Q(t),t)=9-(4/(100-t))*Q(t);

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJRRzYiNiNJInRHRilGKywmIiIqIiIiKiYsJiIkKyJGLkYrISIiRjJGJ0YuISIl

dsolve({Eqn4B,Q(0)=0},Q(t));

Ly1JIlFHNiI2I0kidEdGJSwoIiQrJCIiIkYnISIkKiQsJiEkKyJGKkYnRioiIiUjRisiKCsrKyI=

# Note that t is only valid for 0<=t <=100. Eqn5:=diff(u(t),t)=k*(R-u(t)); #R is the constant environmental temp

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ1RzYiNiNJInRHRilGKyomSSJrR0YpIiIiLCZJIlJHRilGLkYnISIiRi4=

dsolve(Eqn5,u(t));

Ly1JInVHNiI2I0kidEdGJSwmSSJSR0YlIiIiKiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKiZJImtHRiVGKkYnRiohIiJGKkkkX0MxR0YlRipGKg==