Separable Differential Equation Solver Calculator

Enter the constant k, an initial condition y(x0) = y0, and the point where you need the solution evaluated. The calculator separates the variables, identifies the general exponential family, applies the initial condition, and returns a stable numerical value at the requested point. It is useful for checking coursework and for modeling quantities whose instantaneous rate of change is proportional to their current amount.

Recognize the standard separable equation

The calculator is designed for the common equation dy/dx = ky, where k is a fixed real constant. It appears whenever a quantity changes at a rate proportional to its present value: unrestricted population growth, radioactive decay, continuously compounded balances, first-order removal, and idealized cooling differences are familiar examples. The equation is separable because every factor involving y can be moved to one side and every factor involving x to the other, producing dy/y = k dx when y is nonzero. The zero solution is also retained through the initial condition y0 = 0. This focused form makes the answer exact and deterministic; it is not a numerical step approximation. Supply k directly, including a negative value for decay or zero for a constant solution. Then provide x0 and y0 together as the known point on the solution curve. The solver does not accept an arbitrary derivative expression, variable coefficients, forcing terms, or equations that merely resemble this form after a complicated substitution.

From separation to the particular solution

Integrating the separated equation gives ln|y| = kx plus an integration constant. Exponentiating absorbs the sign and positive exponential factor into a single arbitrary constant C, so the general family is y = C exp(kx). Applying y(x0) = y0 determines C = y0 exp(-kx0). The same result is more clearly written as y(x) = y0 exp(k(x - x0)), which is the particular solution reported by this tool. That shifted form is convenient because substituting x0 immediately makes the exponential equal one and recovers y0. It also avoids separately displaying a potentially awkward value of C. If y0 is zero, the formula correctly returns the equilibrium solution y = 0 even though dividing by y during the informal separation step would exclude it. The calculator reports the separation statement, general family, particular formula, and evaluation so you can follow the reasoning instead of receiving an unexplained number. All computations use JavaScript floating-point arithmetic, followed by presentation rounding at your chosen precision.

Evaluate the solution and interpret the result

Choose the target x independently of the initial point; it may be earlier, equal to, or later than x0. The solver computes the dimensionless exponent k(x - x0), evaluates the exponential, multiplies by y0, and rounds the displayed result from zero through fifteen decimal places. A positive k makes the magnitude grow as x increases, while a negative k makes it decay toward zero. Moving backward from x0 reverses those tendencies. The sign of a nonzero solution is inherited from y0 because an exponential factor is always positive, so this model never crosses zero. Check that the units are consistent: if x is measured in hours, k must have units of inverse hours, and the output y keeps the same units as y0. Extremely large exponents can exceed finite numeric range; the calculator rejects those cases rather than returning a misleading infinity. Use the exact formula for analysis and the evaluated value for a specific prediction, comparison, or answer check. For arbitrary separable functions f(x)g(y), additional symbolic integration would be required and is outside this calculator's deliberately bounded scope.