## 3/(1/3)

Input |
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3/(1/3) |

Exact result |

9 |

Number name |

nine |

Visual representation |

* * * * * * * * * |

Category: Differential Equations

Input |
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3/(1/3) |

Exact result |

9 |

Number name |

nine |

Visual representation |

* * * * * * * * * |

Input |
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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, … |

Input interpretation |
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plot | 3 x |

Input |
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3/4×4 |

Exact result |

3 |

Number name |

three |

Visual representation |

* * * |

Input interpretation |
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inflection points | x + sin(x) |

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 |

Derivative |
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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) |

Input interpretation |
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stationary points | sin(x y) |

Result |

sin(x y) = 0 at (x, y) = (0, 0) (saddle point) |

Input interpretation |
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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 |

Input interpretation |
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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 |