Online-Calculator for ordinary linear second order differential equations

The differential equation is given as follows:

y′′ + p(x) y′ + q(x) y = F(x)

with the initial values

  y(x0) = y0   and   y′(x0) = y′0

Numerical solution of the 2.order differential equation

The solution of the differential equation 2.order is calculated numerically. The method can be selected. Three Runge-Kutta methods are available: Heun, Euler and Runge-Kutta 4.Order. The initial values ​​for y0 and y′0 can be varied by pulling the dots in the charts. The value for x0 can be set in numeric input field right. In the text boxes for the functions p, q and F up to three parameters a, b and c can be used which can be varied by means of the slider in the upper graph. In state-space diagram the solutions y1 and y2 of the corresponding first order differential equation system are applied. The diagram shows y2 over y1. The number of grid vectors in state-space diagram can be set in the numeric field for the grid points. In the state-space diagram is plotted y2 on the vertical axis and y1 about the horizontal axis.

Scale:
p(x):
q(x):
F(x):
Steps:
Method:
ODE y:

Initial values

x0=
y0=
y′0=

Parameter values

a=
b=
c=

Axes ranges

x-min=
x-max=
y-min=
y-max=

Parameter ranges

a-min=
b-min=
c-min=

Parameter ranges

a-max=
b-max=
c-max=

Solution in state space (phase space)

Grap the start point to move the initial values. The grid vectors show the initial direction if the ODE starts at this points.

Scale:
Grid points:
Scale=
Curve:
Grid vectors:

Axes ranges

y1-min=
y1-max=
y2-min=
y2-max=

p(x) =

q(x) =

F(x) =

cl
ok
Pos1
End
7
8
9
/
x
4
5
6
*
a
b
c
1
2
3
-
π
(
)
0
.
+
sin
cos
tan
ex
ln
xa
a/x
^
asin
acos
atan
x2
√x
ax
a/(x+b)
|x|
sinh
cosh
a⋅x+c / b⋅x+c
a+x / b+x
x2-a2/ x2+b2
a / x+b
1+√x / 1-√y
exsin(x)cos(x)
x+a
ea⋅x
a⋅x2+b⋅x+c
FunctionDescription
sin(x)Sine of x
cos(x)Cosine of x
tan(x)Tangent of x
asin(x)arcsine
acos(x)arccosine of x
atan(x)arctangent of x
atan2(y, x)Returns the arctangent of the quotient of its arguments.
cosh(x)Hyperbolic cosine of x
sinh(x)Hyperbolic sine of x
pow(a, b)Power ab
sqrt(x)Square root of x
exp(x)e-function
log(x), ln(x)Natural logarithm
log(x, b)Logarithm to base b
log2(x), lb(x)Logarithm to base 2
log10(x), ld(x)Logarithm to base 10
more ...

Transformation

With a substitution the differential equation of second order can be transformed to a differential system of first order.

Substitution:

y1 = y

y2 = y′

So the resulting ODE system of 1.order is:

y1′ = y2

y2′ = F(x) - p(x) y2 - q(x) y1

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