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How do you solve a first order differential equation?

by Lennon Wade

How do you solve a first order differential equation?

A first order differential equation is linear when it can be made to look like this: dy dx + P(x)y = Q(x) Where P(x) and Q(x) are functions of x. To solve it there is a special method: We invent two new functions of x, call them u and v, and say that y=uv.

What is the general solution to a differential equation?

The general solution is simply that solution which you achieve by solving a differential equation in the absence of any initial conditions. The last clause is critical: it is precisely because of the lack of initial conditions that only a general solution can be computed.

How to solve a differential equation?

Solving. So a Differential Equation can be a very natural way of describing something. But it is not very useful as it is.

  • Separation of Variables. All the x terms (including dx) to the other side. If that is the case,you will then have to integrate and simplify the solution.
  • First Order Linear. Where P (x) and Q (x) are functions of x.
  • Homogeneous Equations. There is another special case where Separation of Variables can be used called homogeneous.
  • Bernoulli Equation. When n = 0 the equation can be solved as a First Order Linear Differential Equation.
  • Second Order Equation. In this type of equation the second derivative makes its appearance. There are many distinctive cases among these equations.
  • Undetermined Coefficients. Once we have found the general solution and all the particular solutions,then the final complete solution is found by adding all the solutions together.
  • Variation of Parameters. This is a more general method than Undetermined Coefficients. Finally we complete solution by adding the general solution and the particular solution together.
  • Exact Equations and Integrating Factors
  • Ordinary Differential Equations (ODEs) vs Partial Differential Equations (PDEs) All of the methods so far are known as Ordinary Differential Equations (ODE’s).

    • How do you find the solution set for an equation?

    To find the solution set of an equation with a given domain, you first need to plug each value in the domain into the equation to get the respective range values. Create ordered pairs from these values and write them as a set.