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Contents >> Applied Mathematics >> Numerical Methods >> Ordinary Differential Equations >> Examples

Ordinary differential equations - Example

Example  [1].

Calculate by the Runge-Kutta’s method integral of the differential equationat the initial condition on the segment [0, 0.5] with the step of integration

S o l u t i o n.  Let’s calculate. For this purpose at the first we’ll consistently calculate:

Now we receive:

and, consequently,

The subsequent approaches are similarly calculated. Results of calculations are tabulated:


 Results of numerical integration of the differential
equation (1) by the fourth order Runge-Kutta’s method

          

i

 

          

x

         

y

 

k = 0.1 ( x + y )

         

Δy

0

1

1

1

0.1

 

0.05

1.05

1.1

0.22

 

0.05

1.055

1.105

0.221

 

0.1

1.1105

1.210

0.1210

 

 

 

 

1/60.6620=0.1103

1

0.1

1.1103

1.210

0.1210

 

0.15

1.1708

1.321

0.2642

 

0.15

1.1763

1.326

0.2652

 

0.2

1.2429

1.443

0.1443

 

 

 

 

1/60.7947=0.1324

 2

0.2

1.2427

1.443

0.1443

 

0.25

1.3149

1.565

0.3130

 

0.25

1.3209

1.571

0.3142

 

0.3

1.3998

1.700

0.1700

 

 

 

 

1/60.9415=0.1569

3

0.3

1.3996

1.700

0.1700

 

0.35

1.4846

1.835

0.3670

 

0.35

1.4904

1.840

0.3680

 

0.4

1.5836

1.984

0.1984

 

 

 

 

1/61.1034=0.1840

4

0.4

1.5836

1.984

 0.1984

 

0.45

1.6828

2.133

0.4266

 

0.45

1.6902

2.140

0.4280

 

0.5

1.7976

2.298

0.2298

 

 

 

 

1/61.2828=0.2138

5

0.5

1.7974

 

 

 

Soу (0.5) =1.7974.

For comparison the exact decision of the differential equation (1) is:

whence 

Thus, exact and numerical solutions of the equation (1) have coincided up to the fourth decimal place.

The fourth order Runge-Kutta’s method also is widely applied to the numerical solution of systems of ordinary differential equations.

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