164.872127
8.243606
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13.862944
164.872127
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164.872127
8.243606
20
13.862944
164.872127
0
The Differential Equation Solver computes solutions for four of the most important first-order ordinary differential equations encountered in science and engineering: exponential growth, exponential decay, Newton's law of cooling, and logistic growth. These equations model phenomena ranging from population dynamics and radioactive decay to heat transfer and epidemic spread.
A differential equation relates a function to its derivatives. Rather than solving symbolically (which requires advanced mathematical software), this calculator uses the known closed-form solutions for each equation type. Since each of these four models has an exact analytical solution, the calculator evaluates the solution formula directly at your chosen time $$t$$, delivering precise results instantly.
Exponential growth ($$\frac{dy}{dt} = ky$$, $$k > 0$$) describes unrestricted growth where the rate of change is proportional to the current value. Bacterial colonies in nutrient-rich media, compound interest, and early-stage epidemic spread all follow this pattern. The solution $$y(t) = y_0 e^{kt}$$ grows without bound, with the doubling time given by $$t_d = \frac{\ln 2}{k}$$.
Exponential decay ($$\frac{dy}{dt} = -ky$$, $$k > 0$$) models processes that diminish proportionally, such as radioactive decay, drug clearance from the body, and capacitor discharge. The solution $$y(t) = y_0 e^{-kt}$$ approaches zero asymptotically, with half-life $$t_{1/2} = \frac{\ln 2}{k}$$.
Newton's law of cooling ($$\frac{dT}{dt} = -k(T - T_a)$$) describes how an object's temperature $$T$$ approaches the ambient temperature $$T_a$$ at a rate proportional to the temperature difference. The solution $$T(t) = T_a + (T_0 - T_a)e^{-kt}$$ is used in forensic science (estimating time of death), food safety (cooling hot dishes), and thermal engineering.
Logistic growth ($$\frac{dy}{dt} = ry(1 - y/K)$$) adds a carrying capacity $$K$$ to the growth model, producing an S-shaped (sigmoidal) curve. This equation accurately models population growth with limited resources, adoption of new technologies, and tumor growth in constrained environments. The solution is $$y(t) = \frac{K}{1 + Ae^{-rt}}$$ where $$A = \frac{K - y_0}{y_0}$$.
Select the equation type, enter your parameters and initial condition, then specify the time at which you want to evaluate the solution. The calculator returns the function value, the solution form, the half-life or doubling time, and the instantaneous rate of change at your chosen time.
This calculator evaluates known analytical solutions for four standard first-order ODEs.
Exponential Growth/Decay: The equation $$\frac{dy}{dt} = ky$$ has the general solution:
$$y(t) = y_0 e^{kt}$$
When $$k > 0$$ the function grows; when $$k < 0$$ it decays. The doubling time (growth) or half-life (decay) is:
$$t_{1/2} = \frac{\ln 2}{|k|}$$
Newton's Cooling: The equation $$\frac{dT}{dt} = -k(T - T_a)$$ has solution:
$$T(t) = T_a + (T_0 - T_a)e^{-kt}$$
The half-life of the temperature difference is again $$\frac{\ln 2}{k}$$.
Logistic Growth: The equation $$\frac{dy}{dt} = ry\left(1 - \frac{y}{K}\right)$$ has solution:
$$y(t) = \frac{K}{1 + Ae^{-rt}}, \quad A = \frac{K - y_0}{y_0}$$
The instantaneous rate at time $$t$$ is computed by substituting $$y(t)$$ back into the original ODE: $$\frac{dy}{dt} = ry(t)(1 - y(t)/K)$$.
y(t) is the value of the unknown function at time $$t$$. For growth/decay models this is the quantity (population, mass, concentration); for cooling it is the temperature.
Solution Form shows the analytical formula used. This helps you verify which model is being applied and write down the full solution for your records.
Half-life / Doubling Time indicates how long it takes for the quantity to halve (decay/cooling) or double (growth). For logistic models, this is the characteristic time scale $$\ln 2 / r$$.
Rate dy/dt at t is the instantaneous rate of change at your chosen time, telling you how fast the quantity is currently changing. A positive rate means the quantity is increasing; a negative rate means it is decreasing.
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Starting with 100 units and k = 0.0866 (corresponding to a half-life of about 8 time units), after t = 8 roughly half the material remains: y(8) = 100·e^(-0.0866×8) ≈ 50.03. The half-life is ln(2)/0.0866 ≈ 8.00.
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Starting from 50 organisms with carrying capacity K = 1000 and growth rate r = 0.1, after 40 time units the population reaches about 646. The growth rate at t = 40 is dy/dt ≈ 22.87 — still growing but decelerating as it approaches K = 1000.
This calculator solves four first-order ODEs with known closed-form solutions: exponential growth (dy/dt = ky, k > 0), exponential decay (dy/dt = -ky), Newton's law of cooling (dT/dt = -k(T - Tₐ)), and logistic growth (dy/dt = ry(1 - y/K)). These cover the most common models in science and engineering.
Most differential equations do not have closed-form analytical solutions. This calculator focuses on equations whose exact solutions are known, allowing precise evaluation without numerical approximation. For arbitrary ODEs, numerical methods like Euler's method or Runge-Kutta are required.
They are mathematically identical: both equal ln(2)/|k|. The term 'half-life' is used for decay processes (time for the quantity to halve), while 'doubling time' applies to growth processes (time for the quantity to double). The formula is the same because exponential growth and decay are symmetric transformations.
Newton's law states that the rate of temperature change is proportional to the difference between the object's temperature and the ambient temperature. The proportionality constant k depends on the material and the medium. A hot coffee (T₀ = 90°C) in a room (Tₐ = 20°C) cools rapidly at first, then more slowly as it approaches room temperature.
Carrying capacity K is the maximum sustainable population or quantity that the environment can support. As y approaches K, the growth rate slows and eventually reaches zero. The logistic curve is S-shaped: slow initial growth, rapid middle growth, then deceleration as y nears K.
Mathematically yes — the formulas work with negative y₀. However, in most real-world applications (population, mass, temperature above absolute zero), negative values are not physically meaningful. For exponential and logistic models, positive initial values are standard.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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