First-Order Differential Equation Solver: dy/dx + a y = b x + c

How this calculator works

A first-order linear differential equation has the form dy/dx + a y = f(x). The standard method is the integrating factor: multiply both sides by e^(a x) so that the left side becomes the derivative of y e^(a x), then integrate and divide by e^(a x).

The answer is always a particular solution, which follows the right-hand side, plus a constant times e^(-a x), which is the part that dies away if a is positive. The starting value fixes the constant. For dy/dx + y = x with y(0) = 0, the answer is y = x - 1 + e^(-x).

This solver covers a constant a and a right-hand side of the form b x + c, which comes up in cooling, charging circuits, mixing tanks and simple motion with drag. It shows the integrating factor, the particular solution, the constant, the solution and a graph.

Formulas used

• \mu = e^{ax}
• y_p = \frac{b}{a}x + \frac{c}{a} - \frac{b}{a^2}
• y = y_p + D\,e^{-a(x-x_0)}

Worked example

With the default numbers in the calculator above:

1. The equation and the starting value: dy/dx + 1.00 y = 1.00 x + 0.00
2. Integrating factor: e^(1.00 x)
3. A particular solution (the part that follows the right side): y_p = 1.00 x + (-1.00)
4. Constant from the starting value: 1.00
5. The solution: y = 1.00 x + (-1.00) + (1.00) e^(-1.00 (x - 0.00))
6. Time constant of the part that dies away: 1.00
7. Value of y at the x you chose: 1.14

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: a (the number multiplying y) 2; b (the number multiplying x on the right) 0; c (the constant on the right) 4; x to find y at 1.

1. The equation and the starting value: dy/dx + 2.00 y = 0.00 x + 4.00
2. Integrating factor: e^(2.00 x)
3. A particular solution (the part that follows the right side): y_p = 0.00 x + (2.00)
4. Constant from the starting value: -2.00
5. The solution: y = 0.00 x + (2.00) + (-2.00) e^(-2.00 (x - 0.00))
6. Time constant of the part that dies away: 0.50
7. Value of y at the x you chose: 1.73

Common questions

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Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any university engineering mathematics text (first-order linear ordinary differential equations); Kreyszig, Advanced Engineering Mathematics.

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