Interest as a Differential Equation

These are some rough notes that demonstrates how interest can be expressed as a differential equation. As a bonus, the differential equation is discretized to handle discrete payments.

In[]:=

DSolve[D[B[t],t]==rB[t]–P,B[t],t]

Out[]=

B[t]

P

r

+

rt





1



In[]:=

DSolve[{D[B[t],t]==rB[t]–P,B[0]==B0},B[t],t]//FullSimplify

Out[]=

B[t]

P+

rt



(–P+B0r)

r



In[]:=

eqn=D[P[t],t]rP[t]–M

Out[]=

′

P

[t]–M+rP[t]

In[]:=

soln=DSolve[​​{​​eqn​​},​​P[t],t,DiscreteVariables{t}]

Out[]=

P[t]

M

r

+

rt





1



In[]:=

soln=DSolve[​​{​​eqn/.M0,​​P[0]P0​​},P[t],t]

Out[]=

{{P[t]

rt



P0}}

In[]:=

First@First@soln/.t1//N​​

Out[]=

P[1.]59.343

sol=NDSolve[​​{​​y'[t]a[t],​​WhenEvent[Mod[t,1],a[t]y[t]],y[0]1,a[0]1},{y,a},{t,0,4},DiscreteVariables{a}​​]

Out[]=

yInterpolatingFunction

Domain: {{0.,4.}} Output: scalar

,aInterpolatingFunction

Domain: {{0.,4.}} Output: scalar



In[]:=

Plot[Evaluate[{y[t],a[t]}/.sol],{t,0,4}]

Out[]=