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Category: Differential Equations

3%2

3%2

Input
3 mod 2
Result
1
Number name
one
Visual representation
*
Integers congruent to 1 mod 2
3, 5, 7, 9, 11, 13, 15, 17, 19, 21, …
Find radius of convergence

Find radius of convergence

Input
convergence conditions | sum_n^infinity x^n
Interval of convergence
{x: abs(x)<1}
Inequality plot
| | abs(x)
| 1
Alternate form assuming x is positive
x<1
Alternate form assuming x is real
sqrt(x^2)<1
Solution
-1
Integer solution
x = 0
Find dy/dx by implicit differentiation

Find dy/dx by implicit differentiation

Derivative
d/dx((dy(x))/(dx)) = y”(x)
Differential equation solution
y(x) = c_2 x + c_1
Plots of sample individual solutions
| | | | y(0) = 1
y'(0) = 0
| | | | y(0) = 0
y'(0) = 1
Sample solution family
(sampling y(0) and y'(0))
Possible Lagrangian
(script capital l)(y’, y) = y’^2/2
Series expansion at x=0
y”(0) + x y^(3)(0) + 1/2 x^2 y^(4)(0) + 1/6 x^3 y^(5)(0) + 1/24 x^4 y^(6)(0) + O(x^5)
(Taylor series)
Evaluate the iterated integral

Evaluate the iterated integral

Input interpretation
repeated integral
Alternate name
iterated integral
Definition
A repeated integral is an integral taken multiple times over a single variable (as distinguished from a multiple integral, which consists of a number of integrals taken with respect to different variables). The first fundamental theorem of calculus states that if F(x) = D^(-1) f(x) is the integral of f(x), then
integral_0^x f(t) dt = F(x) – F(0).
Now, if F(0) = 0, then
F(x) = integral f(x) dx = integral_0^x f(t) dt.
Related topics
fractional integral | Fubini theorem | integral | multiple integral
Subject classifications
definite integrals | indefinite integrals
Expanding polynomials

Expanding polynomials

Input interpretation
expand | 3 + x + x^3 + 2 x (1 + x)
Result
x^3 + 2 x^2 + 3 x + 3
Expansion over finite field
GF(2) | x^3 + x + 1
p-adic expansion
p = 3 | (x^3 – x^2) + (x^2 + x + 1) 3