Solving gives \( S(t) = 10e^{-0.0025t} + 20(1 - e^{-0.0025t}) \).

["# Solving the Differential Equation: ( S(t) = 10e^{-0.0025t} + 20(1 - e^{-0.0025t}) )", "When analyzing exponential growth and decay models in science, engineering, and finance, differential equations frequently arise. One such expression that models real-world behavior over time is:", "[\nS(t) = 10e^{-0.0025t} + 20(1 - e^{-0.0025t})\n]", "This article explains how to solve and interpret this function, simplify it to reveal its true form, and understand its practical applications in solving and modeling differential equations.", "---", "## What Is This Function?", "The given expression models a process where an initial quantity stabilizes toward a steady-state value, influenced by decaying and constant components. Let’s break it down:", "[\nS(t) = 10e^{-0.0025t} + 20(1 - e^{-0.0025t})\n]", "At first glance, notice that ( e^{-0.0025t} ) appears with positive and negative coefficients, suggesting simplification.", "---", "## Step 1: Simplify the Expression", "Begin by expanding the second term:", "[\nS(t) = 10e^{-0.0025t} + 20 - 20e^{-0.0025t}\n]", "Now combine the exponential terms:", "[\nS(t) = (10e^{-0.0025t} - 20e^{-0.0025t}) + 20 = -10e^{-0.0025t} + 20\n]", "Wait! This appears incorrect because ( S(t) ) should increase with ( t ), yet the exponential term subtracts here. Let’s recheck:", "Actually:", "[\nS(t) = 10e^{-0.0025t} + 20(1 - e^{-0.0025t}) = 10e^{-0.0025t} + 20 - 20e^{-0.0025t} = (10 - 20)e^{-0.0025t} + 20 = -10e^{-0.0025t} + 20\n]", "But this implies ( S(t) < 20 ) initially (since ( -10e^{-0.0025t} < 0 )), and approaches 20 from below — an unusual decreasing form unless constrained by context.", "Wait — reconsider the interpretation. This function likely arises from solving a linear first-order differential equation modeling systems approaching equilibrium.", "Suppose this $ S(t) $ is the solution to:", "[\n\frac{dS}{dt} = -0.005S(t) + 2 \quad \ ext{with initial condition } S(0) \ ext{ determined by system state}\n]", "Let’s verify:", "Let’s differentiate ( S(t) = -10e^{-0.0025t} + 20 ):", "[\n\frac{dS}{dt} = -10 \cdot (-0.0025)e^{-0.0025t} = 0.025 \cdot e^{-0.0025t}\n]", "But the original differential equation matching this derivative is:", "[\n\frac{dS}{dt} = -0.0025S(t) + 2\n]", "Let’s check if these match.", "Assume ( S(t) = A e^{-0.0025t} + B ), plug into the equation:", "[\n\frac{dS}{dt} = -0.0025A e^{-0.0025t}\n]\n[\n-0.0025S + 2 = -0.0025(A e^{-0.0025t} + B) + 2 = -0.0025A e^{-0.0025t} -0.0025B + 2\n]", "Equate:", "[\n-0.0025A e^{-0.0025t} = -0.0025A e^{-0.0025t} \quad \ ext{(OK)}\n]\n[\n0 = -0.0025B + 2 \Rightarrow B = \frac{2}{0.0025} = 800\n]", "So general solution is:", "[\nS(t) = A e^{-0.0025t} + 800\n]", "Now apply initial condition from problem:", "Given ( S(t) = -10e^{-0.0025t} + 20 ), then comparing,", "[\nA = -10, \quad B = 20 \Rightarrow \ ext{but } B <br/>\ne 800 \ ext{ unless } S(0) = 10e^{0} + 20(1 - 1) = 10\n]", "Wait — this suggests inconsistency unless the original expression is correct as-is.", "Let’s instead assume:", "[\nS(t) = 10e^{-0.0025t} + 20(1 - e^{-0.0025t}) = 10e^{-0.0025t} + 20 - 20e^{-0.0025t} = 20 - 10e^{-0.0025t}\n]", "So:", "[\nS(t) = 20 - 10e^{-0.0025t}\n]", "This is the simplified form.", "---", "## Step 2: Interpret the Differential Equation", "This function is a solution to:", "[\n\frac{dS}{dt} = -kS(t) + r\n]", "Where:\n- ( k = 0.0025 ) (decay/growth rate)\n- ( r = 20 ) (steady-state value)", "This form models systems approaching equilibrium:\n- Exponential decay of deviation from steady-state\n- Rate proportional to current deviation", "The general solution is:", "[\nS(t) = \frac{r}{k} + \left( S(0) - \frac{r}{k} \right)e^{-kt}\n]", "Compare with our simplified form:", "[\nS(t) = 20 - 10e^{-0.0025t} = \frac{20}{0.0025} - 10e^{-0.0025t}\n]", "Since ( \frac{20}{0.0025} = 8000 ), clearly mismatch — so our earlier assumption about ( r ) may not match.", "Wait — reevaluate.", "Suppose instead:", "From the simplification:", "[\nS(t) = 20 - 10e^{-0.0025t}\n]", "Then:", "[\n\frac{dS}{dt} = -10 \cdot (-0.0025) e^{-0.0025t} = 0.025 e^{-0.0025t}\n]", "Plug into ( \frac{dS}{dt} = -k S(t) + r ):", "[\n0.025 e^{-0.0025t} = -k(20 - 10e^{-0.0025t}) + r = -20k + 10k e^{-0.0025t} + r\n]", "Match exponentials:", "Coefficient of ( e^{-0.0025t} ):", "[\n0.025 = 10k \Rightarrow k = 0.0025\n]", "Constant terms:", "[\n0 = -20k + r \Rightarrow r = 20 \cdot 0.0025 = 0.05\n]", "Wait — but earlier we have ( S(t) = 20 - 10e^{-0.0025t} ), so steady-state is 20, but from derivative, ( S(t) \ o 20 ) as ( t \ o \infty ), and decay term gives rate proportional to ( -10e^{-0.0025t} ), so:", "[\n\frac{dS}{dt} = 0.025 e^{-0.0025t} = -0.0025(20 - 10e^{-0.0025t}) + r = -0.05 + 0.025e^{-0.0025t} + r\n]", "Then:", "[\n0.025e^{-0.0025t} = -0.05 + 0.025e^{-0.0025t} + r \Rightarrow 0 = -0.05 + r \Rightarrow r = 0.05\n]", "Contradiction — unless initial conditions or interpretation differ.", "But the function as given:\n[\nS(t) = 10e^{-0.0025t} + 20(1 - e^{-0.0025t}) = 20 - 10e^{-0.0025t}\n]", "Is a well-defined analytical solution — likely representing relaxation toward 20 with time constant ( \ au = \frac{1}{0.0025} = 400 ) units.", "---", "## Step 3: Use This to Analyze Behavior", "- At ( t = 0 ):\n ( S(0) = 20 - 10(1) = 10 )", "- As ( t \ o \infty ): ( e^{-0.0025t} \ o 0 \Rightarrow S(t) \ o 20 )", "- Rate of change:\n ( \frac{dS}{dt} = 0.025 e^{-0.0025t} > 0 ) when ( S(t) < 20 ), so $ S(t) $ increases toward 20.", "This models a process such as:\n- Cooling toward ambient temperature\n- Drug concentration decaying toward steady level\n- Electrolyte balance stabilizing", "---", "## Step 4: Applications of This Model", "Such functions are used in:", "- Pharmacokinetics: Drug absorption/distribution\n- Thermal systems: Temperature equilibration\n- Finance: Interest rate convergence\n- Population dynamics: Growth approaching carrying capacity", "The exponential decay envelope shows how quickly equilibrium is approached — determined by ( k = 0.0025 ) (smaller = slower change).", "---", "## Step 5: Solving Differential Equations via This Solution", "Suppose we are given:", "[\n\frac{dS}{dt} = -0.0025S(t) + 2, \quad S(0) = 10\n]", "This matches our earlier ODE and produces:", "[\nS(t) = 8000 - 7990e^{-0.0025t} \quad ? \quad \ ext{No — too large}\n]", "Wait — contradiction again.", "But our solution is:", "[\nS(t) = 20 - 10e^{-0.0025t}\n]", "So initial value ( S(0) = 10 ), matches.", "Then:", "[\n\frac{dS}{dt} = 0.025 e^{-0.0025t}\n]", "Plug into ODE:", "[\n\ ext{RHS} = -0.0025(20 - 10e^{-0.0025t}) + r = -0.05 + 0.025e^{-0.0025t} + r\n]", "Set equal:", "[\n0.025 e^{-0.0025t} = -0.05 + 0.025 e^{-0.0025t} + r \Rightarrow r = 0.05\n]", "So the correct ODE is:", "[\n\frac{dS}{dt} = -0.0025 S(t) + 0.05, \quad S(0) = 10\n]", "Thus, the function ( S(t) = 20 - 10e^{-0.0025t} ) is a solution only if initial condition is ( S(0) = 10 ), and rate constant recalculated accordingly.", "---", "## Summary", "- The expression ( S(t) = 10e^{-0.0025t} + 20(1 - e^{-0.0025t}) ) simplifies to ( S(t) = 20 - 10e^{-0.0025t} "]