The Linear Operator:
L[y] = y″ + a y′ + b y
Operator & Linearity
Homogeneous IVP — Constant Coefficients
Choose y₁, y₂ (serving as v₁, v₂)
Choose y₁(x)
sin(kx)
cos(kx)
e^{r x}
x
x^2
x^3
Choose the parameter k or r for y₁(x)
Choose y₂(x)
cos(kx)
sin(kx)
e^{r x}
x
x^2
x^3
Choose the parameter k or r for y₂(x)
Define the operator as a 2nd-order linear homogeneous ODE with coefficients a, b
a
b
L[y](x) = y″(x) + a·y′(x) + b·y(x)
Domain & Display
x range
Show y′ and y″ as well
Show comparison: L[αy₁+βy₂] vs αL[y₁]+βL[y₂]
Reset
Linear combination of v₁, v₂
α
β
y(x)
L[y](x)
αL[y₁]+βL[y₂] (comparison)
y′, y″ (gray)
The linear operator you chose is:
Explicit linearity for your choices
Hover over the graph to read values.
Choose coefficient C1 for the general solution
Choose coefficient C2 for the general solution
Choose initial conditions
x₀
y(x₀) =
y′(x₀) =
Range & Steps
x range
Δx (smaller = more accurate)
Reset
Choose constant equation coefficients
a
b
y″ + a·y′ + b·y = 0
General solution: y(x)=c₁·y₁(x)+c₂·y₂(x)
y₁: initial conditions
y₂: initial conditions
Solution for chosen IC: y(x₀)=y₀, y′(x₀)=v₀ (dashed)
L[y] = y″ + a·y′ + b·y
Hover over the graph to read values.