= Solution
<Systole> is ventricular contraction and ejection; <diastole> includes ventricular relaxation and filling. Let $V_a,V_v,V_{LV}$ denote arterial, venous and left-ventricular blood volumes. Let $Q^+$ flow through the aortic valve into the arteries, $Q_c$ through the peripheral circulation, and $Q^-$ through the mitral valve into the ventricle. Conservation of volume gives
$$
\boxed{\dot V_a=Q^+-Q_c,\quad\dot V_v=Q_c-Q^-,\quad\dot V_{LV}=Q^--Q^+.}
$$
The same $Q_c$ leaves arteries and enters veins because the capillary compartment has no changing stored volume in this approximation. Adding the equations conserves total modeled blood volume.
Use constant <vascular compliances> $C_a,C_v$, with $\dot V_a=C_a\dot p_a$ and $\dot V_v=C_v\dot p_v$, and <ventricular elastance> $p_{LV}=E_{LV}(t)(V_{LV}-V_0)$. The <hydraulic resistances> and ideal one-way valves give
$$
Q_c=\frac{p_a-p_v}{R_c},\qquad Q^+=\frac{[p_{LV}-p_a]^+}{R_a},\qquad Q^-=\frac{[p_v-p_{LV}]^+}{R_v},
$$
where $[z]^+=\max(z,0)$. Substitution into the conservation equations gives the two stated arterial/venous pressure equations. Since $V_{LV}-V_0=p_{LV}/E_{LV}$, differentiating this identity gives the third equation. The positive parts encode valve closure against a reverse pressure difference.
During isovolumetric contraction both valves are closed. Let $V^+$ be the end-diastolic ventricular volume, $V_0$ its unstressed reference volume, and $p_a^+$ arterial pressure before ejection. The rapidly raised elastance generates $E_s(V^+-V_0)$, so the aortic valve opens if $\boxed{E_s(V^+-V_0)>p_a^+}$.
During ejection, $R_a/R_c\ll1$ makes the open-valve pressure drop small on the peripheral drainage time scale, so $p_{LV}\approx p_a$. The mitral valve is closed, and adding arterial and ventricular volume balances eliminates $Q^+$. With $p_v\ll p_a$ this gives
$$
R_cC_a\dot p_a+R_c\frac{d(p_a/E_{LV})}{dt}\approx-p_a.
$$
For approximately constant systolic elastance, set $C_s=1/E_s$ and reset time to the start of the slow ejection phase. Then
$$
\boxed{p_a(t)\approx\hat p_a\exp\!\left[-\frac{t}{R_c(C_a+C_s)}\right].}
$$
Here $\hat p_a$ is the matched pressure at that start. If peripheral outflow is negligible during the preceding <rapid artery-ventricle pressure equilibration>, conservation gives the more explicit $\hat p_a=[C_ap_a^++V^+-V_0]/(C_a+C_s)$.
After aortic closure the ventricular volume is fixed at the end-systolic value $V^-$. Relaxation decreases $E_{LV}$ until the mitral valve opens at $p_{LV}=p_v$, hence $\boxed{E_{LV}\sim p_v/(V^--V_0)}$, with $p_v$ the venous filling pressure at opening. During filling the aortic valve is closed, so $R_cC_a\dot p_a=-(p_a-p_v)\approx-p_a$. With $p_a^-$ the arterial pressure at the start of filling and time reset there,
$$
\boxed{p_a(t)\approx p_a^-e^{-t/(R_cC_a)}.}
$$
The change from $C_a+C_s$ to $C_a$ is physical: the ejecting ventricle shares the arterial pressure during systole, whereas a closed aortic valve disconnects it during filling.
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