**1)**Consider the *simplest linear model* of the increment $\displaystyle \Delta$**T** to the Earth’s mean surface temperature arising from a forcing $\displaystyle \Delta$**Q** and a feedback factor, $\displaystyle \lambda$:

**C. **$\displaystyle \frac {d Delta T}{dt}$** +**$\displaystyle \lambda$$\displaystyle \Delta$**T =**$\displaystyle \Delta$**Q**

where **C** is the effective heat capacity of the climate system. Assume a linearly increasing radiative forcing, $\displaystyle \Delta$**Q = b.t, **where t is *the time in years from 1850*.

**a)**Write-down an equation for the equilibrium temperature change that would arise from stabilising forcing at each time **t**.

b) Assuming $\displaystyle \Delta$**T=0**at **t=0**, solve the *simplest linear model* equation to give the dynamical temperature change as a function of **t, C, **$\displaystyle \lambda$ and **b.**

I can do a) and i get $\displaystyle \Delta$$\displaystyle T_{eq}$= b.t/$\displaystyle \lambda$

i try b) and i get $\displaystyle \Delta$$\displaystyle T_{cf}$=A$\displaystyle \exp^{\frac{- \lambda.t}{c}}$

and $\displaystyle T_{pi}$=B.b.t therefore giving me CBb+$\displaystyle \lambda$Bbt=bt

therefore $\displaystyle \Delta T$= A$\displaystyle \exp ^{\frac{- \lambda.t}{c}}$ + $\displaystyle \frac {bt^{2}}{C+\lambda . t}$

and $\displaystyle \Delta T$ =0 and t=0

Therefore A = 0

Therefore i get $\displaystyle \Delta T$ = $\displaystyle \frac {bt^{2}}{C+\lambda . t}$

if any one could shed any light that would be useful