# Online-Calculator for ordinary linear second order differential equations.

y′′ + p(x) y′ + q(x) y = F(x)

p(x) =
 none very small small normal thick very thick xxl xxxl xxxxl red black blue green cl ok Pos1 End 7 8 9 / x 4 5 6 * π ( ) 1 2 3 - a b c 0 . + sin cos tan ex ln log10 asin acos atan x2 √x xy |x| sinh cosh
q(x) =
 none very small small normal thick very thick xxl xxxl xxxxl red black blue green cl ok Pos1 End 7 8 9 / x 4 5 6 * π ( ) 1 2 3 - a b c 0 . + sin cos tan ex ln log10 asin acos atan x2 √x xy |x| sinh cosh
F(x) =
 none very small small normal thick very thick xxl xxxl xxxxl red black blue green cl ok Pos1 End 7 8 9 / x 4 5 6 * π ( ) 1 2 3 - a b c 0 . + sin cos tan ex ln log10 asin acos atan x2 √x xy |x| sinh cosh

Direction field:
Numerics:

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

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

x0=
y0= y′0=

a=
b=
c=

## Solution of the differential equations 2.order

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 y 0 and y′ 0 can be varied by pulling the red and blue dot in the first chart. The value for x 0 can be set in Numericsfeld right. In the text boxes for the functions p, q and F up to three parameters a, b and c are used which can be varied by means of the slider in the upper graph. In phase space diagram the solutions y 1 and y 2 are of the corresponding differential equations 1.order applied. The number of calculated solutions in phase space diagram at different initial values ​​can be set in Numericsfeld under Step size Direction field. In the phase-space diagram is plotted on the vertical axis y2 y1 about the horizontal axis.

## Transformation of the differential equations 2.order in a system 1.order

Substitution:

y1 = y

y2 = y′

So the resulting system of 1.order is:

y1′ = y2

y2′ = F(x) - p(x) y2 - q(x) y1

### Usable expressions in the definition of the functions p, q and F

Constants

NameDescription
LN2Natural logarithm of 2
LN10Natural logarithm of 10
LOG2EBase 2 logarithm of EULER
LOG10EBase 10 logarithm of EULER
PIRatio of the circumference of a circle to its diameter
SQRT1_2Square root of 1/2
SQRT2Square root of 2

Trigonometric Functions

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

Logarithm and Exponential

FunctionDescription
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

FunctionDescription
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 p-th decimal.
V(s)Returns the value of the given element, e.g. sliders and angles.