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How to Solve Linear Ordinary Differential Equations

Differential equations are used in many areas of life, whether we are aware of them or not. For example, when a football player tosses the ball to another player, he evaluates equations to know exactly with how much force he needs to throw it to exactly the right spot. These types of equations play a fundamental role in physics and engineering. Ordinary differential equations are different than partial differential equations in that they have only a single independent variable under consideration.

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Instructions

- 1
  Convert the given equation to the form dy/dx + p(x)y = q(x). This is the standard form of an ordinary linear differential equation.
- 2
  Evaluate the integrating factor u(x) = exp(int(p(x)dx)). The integrating factor, u(x), multiplied by the function p(x), will equal u'(x).
- 3
  Evaluate int(u(x)q(x)dx). This, when the constant C is added to it and divided by the integrating factor u(x), will give you a solution to the differential equation.
- 4
  Write down the form of the general solution. It is given in the form of y=(int(q(x)u(x)dx ) + C)/u(x).
- 5
  Integrate both sides of the given equation. In the example y'=2x + 10, integrating y' will yield y and integrating 2x + 10 gives x^2 + x + C.

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