The differential equation is given as follows:
y′′ + p(x) y′ + q(x) y = F(x)
with the initial values
y(x_{0}) = y_{0} and y′(x_{0}) = y′_{0}
The solution of the differential equation 2.order is calculated numerically. The method can be selected. Three RungeKutta methods are available: Heun, Euler and RungeKutta 4.Order. The initial values for y_{0} and y′_{0} can be varied by pulling the dots in the charts. The value for x_{0} 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 statespace diagram the solutions y_{1} and y_{2} of the corresponding first order differential equation system are applied. The diagram shows y_{2} over y_{1}. The number of grid vectors in statespace diagram can be set in the numeric field for the grid points. In the statespace diagram is plotted y_{2} on the vertical axis and y_{1} about the horizontal axis.
Grap the start point to move the initial values. The grid vectors show the initial direction if the ODE starts at this points.
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With a substitution the differential equation of second order can be transformed to a differential system of first order.
Substitution:
y_{1} = y
y_{2} = y′
So the resulting ODE system of 1.order is:
y_{1}′ = y_{2}
y_{2}′ = F(x)  p(x) y_{2}  q(x) y_{1}
Constants
Name  Description 

LN2  Natural logarithm of 2 
LN10  Natural logarithm of 10 
LOG2E  Base 2 logarithm of EULER 
LOG10E  Base 10 logarithm of EULER 
PI  Ratio of the circumference of a circle to its diameter 
SQRT1_2  Square root of 1/2 
SQRT2  Square root of 2 
Trigonometric Functions
Function  Description 

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 
Logarithm and Exponential
Function  Description 

pow(b, e)  e to the b 
sqrt(x)  Square root of x 
exp(x)  EULER to the x 
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 functions
Function  Description 

ceil(x)  Get smallest integer n with n > x. 
abs(x)  Absolute value of x 
max(a, b, c, ...)  Maximum value of all given values. 
min(a, b, c, ...)  Minimum value of all given values. 
random(max = 1)  Generate a random number between 0 and max. 
round(v)  Returns the value of a number rounded to the nearest integer. 
floor(x)  Returns the biggest integer n with n < x. 
factorial(n)  Calculates n! 
trunc(v, p = 0)  Truncate v after the pth decimal. 
V(s)  Returns the value of the given element, e.g. sliders and angles. 
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Here is a list of of further useful calculators:
Calculator:
ODE first order General first order ODE ODE second order ODESystem 2x2 ODESystem 3x3 Exponential growth Logistic growth Riccati equation Bernoulli equationCalculator for single ODEs:
y'+ay=b y'=fy+gy^n y'+2xy=xe^(x^2) y'+xy=x y'+y=x y'=y^2 y'+y^2=1 y'=(Aya)(Byb)