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Home Differential Equations Equations with Seperable Variables Method For Solving
 
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Method For Solving

Method For Solving A Differential Equation With Separable Variables

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Step 1

Find all points y1 where h(y1) = 0.

For each such point, the constant function y(t) = y1 is a particular solution.

Step 2

Separate the variables by dividing by h(y) and multiplying by dt,

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then integrate both sides of the equation. That is, find antiderivatives of each side.

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so that

K(y) = G(t) + C.

If possible, solve for y as a function of t.

Step 3

The general solution is the family of all solutions found in Steps 1 and 2. It will usually depend on a constant C.

Step 4

If an initial value y(t0) = y0 is given, use it to find the constant C and the particular solution of the initial value problem.
Remark The cases h(y) = 0 and h(y) ≠ 0 must be done separately in Steps 1 and 2, because the division by h(y) in Step 2 cannot be done when h(y) = 0.

 

The general solution of a differential equation dy/dt = g(t), where dy/dt is a function of t alone, is just the indefinite integral

14_differential_equations-12.gif

In this case, C is the familiar constant of integration, which is added to a particular solution. For example, the general solution of the differential equation dy/dt = 1/t is y = ln |1| + C.

In the examples that follow, the constant C appears in a more complicated manner.

Solve the initial value problem

Find the general solution

Home Differential Equations Equations with Seperable Variables Method For Solving

Last Update: 2006-11-17