# Step-by-step calculators:

## Ordinary differential equations

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Calculator solves $$F\left(x,\,y,\,y',\,y'',\dots,y^{\left(n\right)}\right)=0$$ — ordinary differential equations (ODE) different orders, namely:

Separable equations: $$p\left(x\right)\mathrm{d}x=q\left(y\right)\mathrm{d}y$$

Homogeneous equations: $$y'=f\left(k\,x,\;k\,y\right)=f\left(x,\;y\right)$$

Bringing to homogeneous, substitution $$y=z^{\lambda}$$

Linear equations of first order: $$y'+a\left(x\right)\,y=b\left(x\right)$$

Equations of the form: $$y'=f\left(\frac{a_1\,x+b_1\,y+c_1}{a\,x+b\,y+c}\right)$$

Bernoulli differential equation: $$y'+a\left(x\right)\,y=b\left(x\right)\,y^n$$

Riccati differential equation: $$y'+a\left(x\right)\,y+b\left(x\right)\,y^2=c\left(x\right)$$

Equation with total differential: $$P\left(x,\;y\right)\,\mathrm{d}x+Q\left(x,\;y\right)\,\mathrm{d}y=0$$

Finding an integrating factor: $$\mu\cdot P\left(x,\;y\right)\,\mathrm{d}x+\mu\cdot Q\left(x,\;y\right)\,\mathrm{d}y=0$$ — where $$\mu=\mu\left(x\right)$$, $$\mu=\mu\left(y\right)$$ or $$\mu=\mu\left(z\left(x,\,y\right)\right)$$

Grouping under differential $$\mathrm{d}\left(F\left(x,\,y\right)\right)=0$$

Equations not resolved with respect to the derivative: $$F\left(x,\;y,\;y'\right)=0$$ — parameter introduction method $$p\,$$; calculating the total differential; substitution $$\mathrm{d}y=p\,\mathrm{d}x$$; decision regarding $$y'$$

Equations, allowing reduction of order — substitution $$y^{\left(k\right)}=z$$ for equations of form $$F\left(x,\,y^{\left(k\right)},\,y^{\left(k+1\right)},\dots,y^{\left(n\right)}\right)=0$$; substitution $$y'=p\left(y\right)$$ for $$F\left(y,\,y',\,y''\,\dots,y^{\left(n\right)}\right)=0$$; homogeneous equation for y and its derivatives $$y',\,y'',\dots,y^{\left(n\right)}$$; homogeneous relatively $$x$$ and $$y$$ in a generalized sense

Homogeneous and nonhomogeneous linear equations with constant coefficients: $$y^{\left(n\right)}+a_{n-1}\,y^{\left(n-1\right)}+\ldots+a_0\,y=f\left(x\right)$$ — with a special right-hand side

Euler's equation: $$x^n\,y^{\left(n\right)}+a_{n-1}\,x^{n-1}\,y^{\left(n-1\right)}+\ldots+a_{1}\,x\,y'+a_0\,y=0$$

Various substitutions from context of an equation

For first-order equations, the Bernoulli method or variations of an arbitrary constant is used

Trigonometric and hyperbolic transformations

Checking for loss of private solutions

During calculations, calculator independently performs grouping, substitutions or multiplication of an equation, choosing a more suitable solution method in the process

## Indefinite and definite integrals

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Calculator solves $$\displaystyle \int{f\left(x\right)\;\mathrm{d}x=F\left(x\right)+C}$$ — indefinite integral using the following methods and techniques:

Table of basic integrals $$\displaystyle\int{x^n}\;\mathrm{d}x=\dfrac{x^{n+1}}{n+1}+C,\;\left(n\neq-1\right)$$, $$\displaystyle\int{a^x}\;\mathrm{d}x=\dfrac{a^x}{\ln\left(a\right)}+C$$$$\dots$$

Sum rule (difference) $$\displaystyle\int{\left(u\pm v\pm w\right)}\;\mathrm{d}x=\int{u}\;\mathrm{d}x\pm\int{v}\;\mathrm{d}x\pm\int{w}\;\mathrm{d}x$$

Multiplication by constant $$\displaystyle\int{c\,f\left(x\right)}\;\mathrm{d}x=c\int{f\left(x\right)}\;\mathrm{d}x$$

Substitution rule $$\displaystyle\int{f\left(x\right)}\;\mathrm{d}x=\left[\begin{array}{c}x=\varphi\left(t\right)\\\mathrm{d}x=\varphi'\left(t\right)\,\mathrm{d}t\end{array}\right]=\int{f\left(\varphi\left(t\right)\right)\,\varphi'\left(t\right)}\;\mathrm{d}t$$

Integration of rational functions: trigonometric $$\mathrm{R}\left(\sin\left(x\right),\;\cos\left(x\right)\right)$$; hyperbolic $$\mathrm{R}\left(\sinh\left(x\right),\;\cosh\left(x\right)\right)$$; rational fractions $$\dfrac{P_k\left(x\right)}{Q_n\left(x\right)}$$

Methods of undetermined coefficients: factorization of polynomials, linear-fractional irrationality $$\mathrm{R}\left(x,\,\left(\dfrac{a\,x+b}{c\,x+d}\right)^{r_1,\dots,\,r_n}\right)$$, the Ostrogradski method $$\displaystyle\int{\dfrac{P\left(x\right)}{Q\left(x\right)}}=\dfrac{P_2\left(x\right)}{Q_2\left(x\right)}+\int{\dfrac{P_1\left(x\right)}{Q_1\left(x\right)}}$$, containing the root of a square trinomial $$\mathrm{R}\left(x, \sqrt{a\,x^2+b\,x+c}\right)$$, direct methods $$\displaystyle\int{\dfrac{P_n\left(x\right)}{\sqrt{a\,x^2+b\,x+c}}}{\;\mathrm{d}x}$$, $$\displaystyle\int{\dfrac{P_m\left(x\right)}{\left(x-\alpha\right)^n\,\sqrt{a\,x^2+b\,x+c}}}{\;\mathrm{d}x}$$, $$\displaystyle\int{\dfrac{M\,x+N}{\left(x^2+p\,x+q\right)^n\,\sqrt{a\,x^2+b\,x+c}}}{\;\mathrm{d}x}$$

Integration by parts $$\displaystyle\int{u}{\;\mathrm{d}v}=u\,v-\int{v}{\;\mathrm{d}u}$$, trigonometric and hyperbolic substitutions, Euler substitutions, integrals of binomial differential $$\displaystyle\int{x^m\,\left(a\,x^n+b\right)^p}{\;\mathrm{d}x}$$

Product of power functions $$\sin^n\left(x\right)\,\cos^m\left(x\right)$$ and hyperbolic $$\sinh^n\left(x\right)\,\cosh^m\left(x\right)$$

Using known formulas of integration, integration with module, integral functions $$\Gamma\left(s,\,x\right)$$, $$\operatorname{Ei}\left(x\right)$$, $$\operatorname{li}\left(x\right)$$, $$\operatorname{Si}\left(x\right)$$, $$\operatorname{Ci}\left(x\right)$$, $$\operatorname{Shi}\left(x\right)$$, $$\operatorname{Chi}\left(x\right)$$, $$\operatorname{Li_2}\left(x\right)$$, $$\operatorname{S}\left(x\right)$$, $$\operatorname{C}\left(x\right)$$, $$\operatorname{erf}\left(x\right)$$, $$\operatorname{erfi}\left(x\right)$$, grouping under differential $$\displaystyle\int{\mathrm{d}\left(\mathrm{F}\left(x\right)\right)}$$, universal trigonometric/hyperbolic substitution, Euler's formula

Power, logarithmic, trigonometric and hyperbolic transformations

Substitutions, groupings using simplifications

Calculator solves the problem $$\displaystyle\int\limits_{b}^{a}{f\left(x\right)}{\;\mathrm{d}x}$$ — of calculating a definite integral by means of an indefinite, applying the Newton-Leibniz formula, period shortening when integrand is even or odd with symmetrical limits, periodicity

To calculate improper integrals, calculator considers limits at infinity, left-sided and right-sided limits at the points of discontinuity of the function on the interval

List of involved math functions:

$$\ln$$ $$\sin$$ $$\cos$$ $$\tan$$ $$\cot$$ $$\arctan$$ $$\arcsin$$ $$\arccos$$ $$\operatorname{arccot}$$ $$\sinh$$ $$\cosh$$ $$\tanh$$ $$\coth$$ $$\operatorname{sech}$$ $$\operatorname{csch}$$ $$\operatorname{arsinh}$$ $$\operatorname{arcosh}$$ $$\operatorname{artanh}$$ $$\operatorname{arcoth}$$ $$\operatorname{arcsec}$$ $$\operatorname{arccsc}$$ $$\operatorname{arsech}$$ $$\operatorname{arcsch}$$ $$\sec$$ $$\csc$$ $$\left|f\right|$$

## Equations

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Calculator solves $$f\left(x\right)=0$$ — equations, namely:

Defines region of admissible values $$D\left(f\right)$$

Linear equations $$a\,x+b=0$$

Quadratic equations with real and complex coefficients $$a\,x^2+b\,x+c=0$$

Reciprocal equations of the 3rd degree $$a\,x^3+b\,x^2+b\,x+a=0$$

Cubic equations $$a\,x^3+b\,x^2+c\,x+d=0$$

Reciprocal equations of the 4th degree $$a\,x^4+b\,x^3+c\,x^2\pm b\,x+a=0$$

Generalized reciprocal equations of the 4th degree $$a\,x^4+b\,x^3+c\,x^2+d\,x+\dfrac{a\,d^2}{b^2}=0$$

Product of four terms of an arithmetic progression $$\left(a\,x+b\right)\,\left(a\,x+b+c\right)\,\left(a\,x+b+2\,c\right)\,\left(a\,x+b+3\,c\right)=d$$

Equations of various powers, logarithmic, trigonometric, hyperbolic and their inverses

Applies the Ferrari method, solving the cubic resolvent for the equation $$a\,x^4+b\,x^3+c\,x^2+d\,x+e=0$$

Finding a rational root $$x=\dfrac{m}{n}$$, factorization $$f_1\left(x\right)\cdots f_n\left(x\right)=0$$

Tabular formulas for trigonometric, hyperbolic and inverse functions

Extracting the root of a complex number $$\sqrt[n]{a+i\,b}$$

Trigonometric and hyperbolic formulas and transformations

Universal trigonometric substitution $$u=\tan\left(\dfrac{x}{2}\right)$$

Binomial theorem $$(a+b)^n=a^n+C^1_n\,a^{n-1}\,b+\ldots+C^{n-1}_n\,a\,b^{n-1}+b^n$$

Sum and difference formulas $$x^n+y^n$$, $$x^n-y^n$$

Grouping terms, taking out a common factor, dividing and multiplying both sides of a equation

Proportions method $$\dfrac{a}{b}=\dfrac{c}{d}\;\Rightarrow\;a\,d=b\,c$$, selection of a full square $$(a+b)^2+c$$

Logarithm of both sides of the equation, exponentiation

Complex logarithm $$\ln\left(a+i\,b\right)$$, Euler formula $$e^{i\,x}=\cos\left(x\right)+i\,\sin\left(x\right)$$

Substitutions from equation context

Transition to a simple functional equation $$f\left(g\left(x\right)\right) = f\left(r\left(x\right)\right)\;\Rightarrow\;g\left(x\right)=r\left(x\right)$$

Substitution of a previously calculated equation into a current equation, search for a solution from RAV values

## Derivative of a function

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After inputting a function $$f\left(x\right)$$ or $$f\left(x,\,y,\,y',\dots,\,z,\,z',\dots\right)$$ — where $$y=y\left(x\right)$$, $$z=z\left(x\right)$$ calculator displays its derivative, along with rules used on concrete steps

The following rules are defined:

Table functions $$\sin\left(x\right)$$, $$\cos\left(x\right)$$$$\,\ldots$$, addition $$u+v$$, subtraction $$u-v$$, multiplication $$u\,v$$, division $$\dfrac{u}{v}$$, various complex functions $$e^{\cos\left(x\right)}$$, power functions $$x^a$$, $$a^x$$, module $$\left|f\right|$$ and sign function $$\operatorname{sgn}\left(f\right)$$

## Limit of a function

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Calculator finds the limit of a function $$\displaystyle\lim_{x\to{a}}{f\left(x\right)}$$, using properties sum $$\displaystyle\lim_{x\to{a}}{f\left(x\right)+g\left(x\right)}=\lim_{x\to{a}}{f\left(x\right)}+\lim_{x\to{a}}{g\left(x\right)}$$, product $$\displaystyle\lim_{x\to{a}}{f\left(x\right)\,g\left(x\right)}=\lim_{x\to{a}}{f\left(x\right)}\,\lim_{x\to{a}}{g\left(x\right)}$$, exponential function $$\displaystyle\lim_{x\to{a}}{a^{f\left(x\right)}}=a^{\lim_{x\to{a}}{f\left(x\right)}}$$ of limits, common limits $$\displaystyle\lim_{x\to{0}}{\dfrac{\sin\left(x\right)}{x}}$$ and $$\displaystyle\lim_{x\to{\infty}}{(1+\dfrac{1}{x})^x}$$, squeeze theorem $$g\left(x\right)\leq f\left(x\right)\leq h\left(x\right)$$, factorization, conjugate multiplication $$\left(a-b\right)\,\left(a+b\right)$$, L'Hôpital's rule $$\displaystyle\lim_{x\to{a}}{\dfrac{f'\left(x\right)}{g'\left(x\right)}}$$, Taylor expansion $$\sum\limits_{n=0}^{\infty}\dfrac{f^{\left(n\right)}\left(a\right)}{n!}\,\left(x-a\right)^n$$, substitutions, groupings and Euler's formula. Calculated both two-sided $$x\to{a}$$, and one-sided limits $$x\to{a+}$$

## Complex numbers

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Calculator converts a complex number $$z$$ to algebraic $$z=a+i\,b$$, trigonometric $$z=r\cdot(\cos(\varphi)+i\,\sin(\varphi))$$ or exponential form $$z=r\,e^{i\,\varphi}$$. Using operations of module $$r=\left|a+i\,b\right|=\sqrt{a^2+b^2}$$, multiplying a fraction by its conjugate $$\dfrac{z}{a+i\,b}\;\Rightarrow\;\dfrac{z\cdot\left(a-i\,b\right)}{\left(a+i\,b\right)\cdot\left(a-i\,b\right)}\;\Rightarrow\;\dfrac{z\cdot\left(a-i\,b\right)}{a^2+b^2}$$, root extraction $$\sqrt[n]{z}=\sqrt[n]{r}\,\left(\cos\left(\dfrac{\varphi+2\,\pi\,\mathrm{k}}{n}\right)+i\,\sin\left(\dfrac{\varphi+2\,\pi\,\mathrm{k}}{n}\right)\right)$$, raising to a power $$z^n=r^n\,\left(\cos\left(n\,\varphi\right)+i\,\sin\left(n\,\varphi\right)\right)$$, formulas for the complex logarithm $$\operatorname{Ln}\left(z\right)=\ln\left(r\right)+i\,(\varphi+2\,\pi\,\mathrm{k})$$, trigonometric $$\sin\left(\alpha\pm\beta\right)=\sin\left(\alpha\right)\,\cos\left(\beta\right)\pm\cos\left(\alpha\right)\,\sin\left(\beta\right)$$, and hyperbolic $$\sinh\left(i\,b\right)=i\,\sin\left(b\right)$$ formulas, as well as Euler's formula $$e^{i\,\varphi}=\cos\left(\varphi\right)+i\,\sin\left(\varphi\right)$$

## Matrix calculations

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Calculator is focused on step by step operations with matrices $$\mathrm{A}$$, $$\mathrm{B}$$ and $$\mathrm{C}$$

Its functionality includes such matrix operations as: addition $$\mathrm{A}+\mathrm{B}$$, multiplication $$\mathrm{C}\cdot\mathrm{B}$$, determinant $$\left|\mathrm{A}\right|$$, transposition $$\mathrm{B}^{\mathrm{T}}$$, rank $$\operatorname{rank}\mathrm{C}$$, inverse matrix $$\mathrm{A}^{-1}$$, exponentiation $$\mathrm{B}^4$$, triangular form $${\scriptsize\left(\begin{matrix}2&3\\0&5\end{matrix}\right)}$$

Multiplying a matrix by a constant (any function) $$a\cdot\mathrm{B}$$ or addition with a constant $$c+\mathrm{A}$$

Calculating the derivative of matrix elements $$\left(\mathrm{C}\right)'_x={\scriptsize\left(\begin{matrix}\left(\mathrm{a_{11}}\right)'_x&\left(\mathrm{a_{12}}\right)'_x\\\left(\mathrm{a_{21}}\right)'_x&\left(\mathrm{a_{22}}\right)'_x\end{matrix}\right)}$$, and similarly, integration of a matrix $$\int{\mathrm{A}}{\;\mathrm{d}x}={\scriptsize\left(\begin{matrix}\int{\mathrm{a_{11}}}{\;\mathrm{d}x}&\int{\mathrm{a_{12}}}{\;\mathrm{d}x}\\\int{\mathrm{a_{21}}}{\;\mathrm{d}x}&\int{\mathrm{a_{22}}}{\;\mathrm{d}x}\end{matrix}\right)}$$

Element-wise applying to a matrix of mathematical functions $$\sin$$, $$\cos$$$$\,\ldots$$ — $$\ln\left(\mathrm{A}\right)={\scriptsize\left(\begin{matrix}\ln\left(\mathrm{a_{11}}\right)&\ln\left(\mathrm{a_{12}}\right)\\\ln\left(\mathrm{a_{21}}\right)&\ln\left(\mathrm{a_{22}}\right)\end{matrix}\right)}$$

Calculator handles both numerical values and combinations of arithmetic operations and functions

If, during a solution, a matrix or a pair of matrices does not satisfy a condition of a current operation, all previously calculated steps are displayed and a discrepancy is clearly indicated

When hovering over calculated elements, all values used in calculation are highlighted. For example, when multiplying matrices, you can see which elements of the row and column are involved in the calculation

All non-matrix operations are performed in the usual order during calculations