## 1st order differencial equasion

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$$\displaystyle \DeltaT =\displaystyle \DeltaQ where C is the effective heat capacity of the climate system. Assume a linearly increasing radiative forcing, \displaystyle \DeltaQ = b.t, where t is the time in years from 1850. a)Write-down an equation for the equilibrium temperature change that would arise from stabilising forcing at each time t. b) Assuming \displaystyle \DeltaT=0at t=0, solve the simplest linear model equation to give the dynamical temperature change as a function of t, C, \displaystyle \lambda and b. I can do a) and i get \displaystyle \Delta$$\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