## 1st order differencial equasion

1)Consider the simplest linear model of the increment $\Delta$T to the Earth’s mean surface temperature arising from a forcing $\Delta$Q and a feedback factor, $\lambda$:
C. $\frac {d Delta T}{dt}$ + $\lambda$ $\Delta$T = $\Delta$Q
where C is the effective heat capacity of the climate system. Assume a linearly increasing radiative forcing, $\Delta$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 $\Delta$T=0at t=0, solve the simplest linear model equation to give the dynamical temperature change as a function of t, C, $\lambda$ and b.

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

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

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

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

and $\Delta T$ =0 and t=0
Therefore A = 0

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

if any one could shed any light that would be useful