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Integrating Factor Calculator

Enter the constant coefficient P and a value of x to compute the integrating factor.

How it is calculated

For the first-order linear equation y′ + P·y = Q(x) the integrating factor is μ(x) = e^(∫P dx). This tool handles the case where P is a constant number p, so the integral is simply p·x and μ = e^(p·x). You enter p and the point x. The tool prints the exponent p·x to four decimals and μ to six decimals. The right-hand side Q(x) is not needed, because μ depends only on P. The constant of integration is left out, which is allowed: any constant only multiplies μ by a number and the final solution is the same. A very large exponent overflows and shows ∞ for μ, a very negative one shows 0.000000.

Worked example

Take the equation y′ + 3y = Q(x), so p = 3, and evaluate the factor at x = 0.5. Exponent: p·x = 3 × 0.5 = 1.5000. Integrating factor: μ = e^1.5 = 4.481689. Multiplying the whole equation by μ(x) = e^(3x) turns the left side into the single derivative (e^(3x)·y)′, which is the reason the method works. With p = −0.5 and x = 4, the exponent is −2.0000 and μ = 0.135335.

Try this example →

Common mistakes

  • Entering the whole coefficient of y when the equation is not in standard form: first divide through so y′ has coefficient 1, because P is the coefficient of y after that division.
  • Getting the sign of p wrong; for y′ − 2y = Q the value to enter is −2, which gives a decaying factor, not 2.
  • Using the tool when P depends on x (for example P = 1/x or P = 2x): the tool assumes a constant p, so the true factor, e.g. x or e^(x²), will not match.
  • Expecting a solution y(x); the tool gives only the factor μ at one point, and you still have to integrate μ·Q yourself.
  • Worrying that a missing constant of integration changes the answer; it does not, and ∞ or 0.000000 outputs simply mean the exponent is too large or small for the display.

About this calculator

For a first-order linear ODE of the form dy/dx + Py = Q, multiplying both sides by the integrating factor μ = e^(∫P dx) turns the left-hand side into the exact derivative d/dx(μy), which can then be integrated directly to solve for y. When P is constant, the integral simplifies to μ = e^(Px), evaluated at whatever x value you enter.

This method — sometimes credited to Leibniz — is the standard technique taught in introductory differential-equations courses for solving linear (but not necessarily separable) first-order equations, and it underlies models in circuit analysis (RC/RL charging curves), population growth with a constant harvesting rate, and Newton's law of cooling. Students and engineers use this calculator to check the integrating factor for a given constant coefficient before carrying out the rest of the solution by hand.

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Comments (1)

  • Editorial team

    This page handles only a constant coefficient p, where the integral is simply p times x. If your equation has P that changes with x, such as 2/x or cos x, you must integrate P yourself first, and the exponent here will not match. A quick check: the factor is always positive, and it equals 1 when x is 0. Which P(x) forms would you like us to support next?

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