The Bernoulli differential equation is a special form of the first order nonlinear differential equation and is:
y′(x) = f(x) ⋅ y(x) + g(x) ⋅ y^{n}(x)
with the initial values
y(x_{0}) = y_{0}
where y is a sought function of x and f(x) and g(x) are continuous functions on an interval I and n is a real number not equal to one. The Bernoulli differential equation occurs frequently in physics and engineering, especially in fluid mechanics and aerodynamics.
The solution of the Bernoulli differential equation is solved numerically. The used method can be selected. Three Runge-Kutta methods are available: Heun, Euler and RK4. The initial value can be varied by dragging the red point on the solution curve. In the input fields for the functions f and g, up to three parameters a, b and c are used which can be varied by means of the slider in the graphics.
f(x) =
g(x) =
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 |
pow(a, b) | Power a^{b} |
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 |
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ODE first order General first order ODE ODE second order ODE-System 2x2 ODE-System 3x3 Exponential growth Logistic growth Riccati 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'=(Ay-a)(By-b)