# How to simplify an equation?

• Aug 27th 2013, 05:55 PM
tabibito
How to simplify an equation?
Hello,
I have the following problem. There are known dependencies between variables.

$a = a1 + a2.$

$C = C1 + C2.$

$C = \frac{a}{{{{\left( {1 + d} \right)}^t}}},C1 = \frac{{a1}}{{{{\left( {1 + d1} \right)}^t}}},C2 = \frac{{a2}}{{{{\left( {1 + d2} \right)}^t}}}.$

The desired quantity is d. In this simple case it can be easily expressed using C1, C2, C, d1, d2. The solution is following

${\left( {1 + d} \right)^t} = \frac{a}{C} = \frac{{a1 + a2}}{C} = \frac{{{{\left( {1 + d1} \right)}^t}C1 + {{\left( {1 + d2} \right)}^t}C2}}{C}.$

$d = {\left( {\frac{{{{\left( {1 + d1} \right)}^t}C1 + {{\left( {1 + d2} \right)}^t}C2}}{C}} \right)^{\frac{1}{t}}} - 1.$

But I am confused with the similar, but a bit more complex problem.
The known dependencies between variables are following

${a_t} = a{1_t} + a{2_t}.$

${C_t} = C{1_t} + C{2_t}.$

${C_{t - 1}} = \frac{{{a_t}}}{{{{\left( {1 + {d_t}} \right)}^t}}} + \frac{{{C_t}}}{{\left( {1 + {d_t}} \right)}},C1 = \frac{{a{1_t}}}{{{{\left( {1 + d{1_t}} \right)}^t}}} + \frac{{C{1_t}}}{{\left( {1 + d{1_t}} \right)}},C2 = \frac{{a2{}_t}}{{{{\left( {1 + d{2_t}} \right)}^t}}} + \frac{{C{2_t}}}{{\left( {1 + d{2_t}} \right)}}.$

t - time index.

The desired quantity is ${d_t}$ .

The goal is to express ${d_t}$

using C1, C2, C, d1, d2, a1 and a2. The solution of the equation seems cumbersome. Is it possible to reduce the resulting formula to elegant form, which would resemble $d = {\left( {\frac{{{{\left( {1 + d1} \right)}^t}C1 + {{\left( {1 + d2} \right)}^t}C2}}{C}} \right)^{\frac{1}{t}}} - 1$?
• Aug 27th 2013, 08:50 PM
chiro
Re: How to simplify an equation?
Hey tabibito.

Does t have a range (i.e. is it say greater than zero)?

If it does, then expand everything out in terms of C1_0 and C2_0 and given these expressions see if you can use a simplification much like the one you used above.

Also does d_t depend on the time t or is constant across all values of time?
• Aug 27th 2013, 09:37 PM
tabibito
Re: How to simplify an equation?
Thank you for the reply, Chiro. t>0. It can be a fraction, say, 1/12 or 1/2 etc.

d_t depends on C1_t-1, C2_t-1, C_t-1, d1_t, d2_t, a1_t and a2_t. As these variables are time-dependent, d_t will also change with time.

I tried replacements, but it seems that the equation resists simplification.

${C_{t - 1}} = \frac{{{a_t} + {C_t}{{\left( {1 + {d_t}} \right)}^{t - 1}}}}{{{{\left( {1 + {d_t}} \right)}^t}}}.$

${\left( {1 + {d_t}} \right)^t} = \frac{{{a_t} + {C_t}{{\left( {1 + {d_t}} \right)}^{t - 1}}}}{{{C_{t - 1}}}}.$

$a{1_t} = C{1_{t - 1}}{\left( {1 + d{1_t}} \right)^t} - \frac{{C{1_t}{{\left( {1 + d{1_t}} \right)}^t}}}{{\left( {1 + d{1_t}} \right)}}.$

${\left( {1 + {d_t}} \right)^t} = \frac{{C{1_{t - 1}}{{\left( {1 + d{1_t}} \right)}^t} - \frac{{C{1_t}{{\left( {1 + d{1_t}} \right)}^t}}}{{\left( {1 + d{1_t}} \right)}} + C{2_{t - 1}}{{\left( {1 + d{2_t}} \right)}^t} - \frac{{C{2_t}{{\left( {1 + d{2_t}} \right)}^t}}}{{\left( {1 + d{2_t}} \right)}} + {C_t}{{\left( {1 + {d_t}} \right)}^{t - 1}}}}{{{C_{t - 1}}}}.$
• Aug 27th 2013, 10:49 PM
tabibito
Re: How to simplify an equation?
Sorry. I mistyped in the first post

It should be

${C_{t - 1}} = \frac{{{a_t}}}{{{{\left( {1 + {d_t}} \right)}^t}}} + \frac{{{C_t}}}{{\left( {1 + {d_t}} \right)}},C{1_{t - 1}} = \frac{{a{1_t}}}{{{{\left( {1 + d{1_t}} \right)}^t}}} + \frac{{C{1_t}}}{{\left( {1 + d{1_t}} \right)}},C{2_{t - 1}} = \frac{{a2{}_t}}{{{{\left( {1 + d{2_t}} \right)}^t}}} + \frac{{C{2_t}}}{{\left( {1 + d{2_t}} \right)}}.$