Question 10.7: Verify Eqs. (10.18) and (10.19)....

Apply the integrating factor \mathrm{exp}(\lambda_{X}t) to Eq. (10.17)

\left[\frac{d}{d t}X(t)+\lambda_{X}X(t)\right]\exp(\lambda_{X}t)=\frac{d}{d t}\{X(t)\exp(\lambda_{X}t)\}=\lambda_{I}I_{o}\exp(-\lambda_{I}t)\exp(\lambda_{X}t)\}

Integrate between 0 and t:

X(t)\exp(\lambda_{X}t)-X_{o}=\lambda_{I}I_{o}{\int_0^t}\ \exp[(\lambda_{X}-\lambda_{I})t^{\prime}]d t^{\prime}={\frac{\lambda_{I}}{\lambda_{X}-\lambda_{I}}}I_{o}(e^{(\lambda_{X}-\lambda_{I})t}-1)

Solving for X:

X(t)=X_{o}e^{-\lambda_{X}t}+{\frac{\lambda_{I}}{\lambda_{X}-\lambda_{I}}}I_{o}(e^{-\lambda_{I}t}-e^{-\lambda_{X}t})

or clearing minus signs

X(t)=X_{o}e^{-\lambda_X t}+\frac{\lambda_{I}}{\lambda_{I}-\lambda_{X}}I_{o}(e^{-\lambda_X{t}}-e^{-\lambda_{I}t})

Substituting in Eqs. (10.14) and (10.15) we have the second required equation:

X(t)=\overline{{{\Sigma}}}_{f}\phi\left[\frac{(\gamma_{I}+\gamma_{X})}{\lambda_{X}+\sigma_{a X}\phi}e^{-\lambda_{X}t}+\frac{\gamma_{I}}{\lambda_{I}-\lambda_{X}}I_{o}(e^{-\lambda_{X}t}-e^{-\lambda_{I}t})\right]