Calculator

# 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 (ODEs) of different orders, namely:

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

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

First-order linear equations: $$y'+a\left(x\right)\,y=b\left(x\right)$$

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

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

Inexact differential equations: $$\mu\cdot P\left(x,\;y\right)\,\mathrm{d}x+\mu\cdot Q\left(x,\;y\right)\,\mathrm{d}y=0$$ — where $$\mu$$ is an integrating factor

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

Equations not solved with respect to the derivative: $$F\left(x,\;y,\;y'\right)=0$$

Equations of the form: $$F\left(x,\,y^{\left(k\right)},\,y^{\left(k+1\right)},\dots,y^{\left(n\right)}\right)=0$$ and $$F\left(y,\,y',\,y''\,\dots,y^{\left(n\right)}\right)=0$$

Linear differential equation with constant coefficients: $$y^{\left(n\right)}+a_{n-1}\,y^{\left(n-1\right)}+\ldots+a_0\,y=f\left(x\right)$$

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

Solves systems of ordinary differential equations:

Linear homogeneous with constant coefficients: $$X'\left(t\right)=A\,X\left(t\right)$$

Linear nonhomogeneous with constant coefficients: $$X'\left(t\right)=A\,X\left(t\right)+f\left(t\right)$$

Solves equations and systems with initial conditions (Cauchy problem)

## 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 integrals using the following methods:

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

Constant multiple rule $$\displaystyle\int{c\,f\left(x\right)}\;\mathrm{d}x=c\int{f\left(x\right)}\;\mathrm{d}x$$

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 $$\mathrm{R}\left(\sinh\left(x\right),\;\cosh\left(x\right)\right)$$; fractions $$\dfrac{P_k\left(x\right)}{Q_n\left(x\right)}$$

Partial fraction decomposition: factorization of polynomials, the Ostrogradsky's method $$\displaystyle\int{\dfrac{P\left(x\right)}{Q\left(x\right)}}=\dfrac{P_2\left(x\right)}{Q_2\left(x\right)}+\int{\dfrac{P_1\left(x\right)}{Q_1\left(x\right)}}$$

Integrals of the form: $$\displaystyle\int\mathrm{R}\left(x,\,\left(\dfrac{a\,x+b}{c\,x+d}\right)^{r_1,\dots,\,r_n}\right)\;\mathrm{d}x$$, $$\displaystyle\int\dfrac{P_n\left(x\right)}{\sqrt{a\,x^2+b\,x+c}}\;\mathrm{d}x$$, $$\displaystyle\int{\dfrac{P_m\left(x\right)}{\left(x-\alpha\right)^n\,\sqrt{a\,x^2+b\,x+c}}}{\;\mathrm{d}x}$$, $$\displaystyle\int{\dfrac{M\,x+N}{\left(x^2+p\,x+q\right)^n\,\sqrt{a\,x^2+b\,x+c}}}{\;\mathrm{d}x}$$, $$\displaystyle\int{x^m\,\left(a\,x^n+b\right)^p}{\;\mathrm{d}x}$$

Integration by parts $$\displaystyle\int{u}{\;\mathrm{d}v}=u\,v-\int{v}{\;\mathrm{d}u}$$

Euler substitution for $$\displaystyle\int\mathrm{R}\left(x, \sqrt{a\,x^2+b\,x+c}\right)\;\mathrm{d}x$$

Uses known formulas of integration, integral of absolute value, integral functions $$\Gamma\left(s,\,x\right)$$, $$\operatorname{Ei}\left(x\right)$$, $$\operatorname{li}\left(x\right)$$, $$\operatorname{Si}\left(x\right)$$, $$\operatorname{Ci}\left(x\right)$$, $$\operatorname{Shi}\left(x\right)$$, $$\operatorname{Chi}\left(x\right)$$, $$\operatorname{Li_2}\left(x\right)$$, $$\operatorname{S}\left(x\right)$$, $$\operatorname{C}\left(x\right)$$, $$\operatorname{erf}\left(x\right)$$, $$\operatorname{erfi}\left(x\right)$$, total differential $$\displaystyle\int{\mathrm{d}\left(\mathrm{F}\left(x\right)\right)}$$, tangent half-angle substitution, Euler's formula $$e^{i\,x}=\cos(x)+i\,\sin(x)$$

Uses exponential, logarithmic, trigonometric, and hyperbolic formulas

Calculator solves $$\displaystyle\int\limits_{b}^{a}{f\left(x\right)}{\;\mathrm{d}x}$$ — definite integrals by applying the fundamental theorem of calculus, checks whether a function is even, odd or periodic

To calculate improper integrals, calculator considers limits at infinity, left-sided and right-sided limits

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 the domain of a function $$\mathrm{dom}\left(f\right)$$

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

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

Cubic equations of the form $$a\,x^3+b\,x^2+b\,x+a=0$$

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

Quartic equations of the form $$a\,x^4+b\,x^3+c\,x^2\pm b\,x+a=0$$ and $$a\,x^4+b\,x^3+c\,x^2+d\,x+\dfrac{a\,d^2}{b^2}=0$$

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

Various exponential, logarithmic, trigonometric, hyperbolic, and inverse to them equations

Applies Ferrari method to solve quartic equations $$a\,x^4+b\,x^3+c\,x^2+d\,x+e=0$$

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

Known solutions of simple trigonometric, hyperbolic and inverse equations

Finding roots of a complex number $$\sqrt[n]{a+i\,b}$$

Substitution by tangent of half angle $$\sin(x)=\dfrac{2\,t}{1+t^2}$$ and $$\cos(x)=\dfrac{1-t^2}{1+t^2}$$ where $$t=\tan\left(\dfrac{x}{2}\right)$$

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

Polynomial identities of sum and difference $$x^n+y^n$$, $$x^n-y^n$$

Combining like terms, factoring out a common term $$x^2+x\;\Rightarrow\; x\,(x+1)$$

Multiplying fractions crosswise $$\dfrac{a}{b}=\dfrac{c}{d}\;\Rightarrow\;a\,d=b\,c$$, completing the square $$(a+b)^2+c$$

Exponentiating both sides to eliminate natural logarithms

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

Simple functional equations $$f\left(g\left(x\right)\right) = f\left(r\left(x\right)\right)\;\Rightarrow\;g\left(x\right)=r\left(x\right)$$

## Derivative of a function

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Calculator computes the derivative of a function $$f\left(x\right)$$ or $$f\left(x,\,y,\,y',\dots,\,z,\,z',\dots\right)$$ and displays rules used to calculate the derivative

The following rules are defined:

Common derivatives of $$x$$, $$\sin(x)$$, $$\cos(x)$$, $$\tan(x)$$, $$\cot(x)$$, $$e^x$$, $$a^x$$, $$\ln(x)$$$$\,\ldots$$

Constant rule $$(c)'=0$$

Constant multiple rule $$\left(c\,f(x)\right)'=c\,f'(x)$$

Sum rule $$\left(f(x)+g(x)\right)'=f'(x)+g'(x)$$

Difference rule $$\left(f(x)-g(x)\right)'=f'(x)-g'(x)$$

Power rule $$\left(x^n\right)'=n\,x^{n-1}$$

Product rule $$\left(f(x)\,g(x)\right)'=f(x)\,g'(x)+g(x)\,f'(x)$$

Quotient rule $$\left(\dfrac{f(x)}{g(x)}\right)'=\dfrac{g(x)\,f'(x)-f(x)\,g'(x)}{\left(g(x)\right)^2}$$

Reciprocal rule $$\left(\dfrac{1}{f(x)}\right)'=\dfrac{-f'(x)}{\left(f(x)\right)^2}$$

Chain rule $$\left(f\left(g(x)\right)\right)'=f'_g\left(g\right)\,g'(x)$$

Absolute value $$\left(\left|x\right|\right)'=\dfrac{x}{\left|x\right|}$$

Sign function $$\left(\operatorname{sgn}\left(f\right)\right)'=2\,\delta\left(x\right)$$ where $$\delta$$ is the Dirac delta function

## 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 the following properties:

Limit of a constant $$\displaystyle\lim_{x\to{a}}C=C$$

Constant multiple rule $$\displaystyle\lim_{x\to{a}}k\,f(x)=k\,\lim_{x\to{a}}f(x)$$

Sum and difference rule $$\displaystyle\lim_{x\to{a}}{f\left(x\right)\pm g\left(x\right)}=\lim_{x\to{a}}{f\left(x\right)}\pm\lim_{x\to{a}}{g\left(x\right)}$$

Product rule $$\displaystyle\lim_{x\to{a}}{f\left(x\right)\,g\left(x\right)}=\lim_{x\to{a}}{f\left(x\right)}\,\lim_{x\to{a}}{g\left(x\right)}$$

Quotient rule $$\displaystyle\lim_{x\to{a}}\dfrac{f(x)}{g(x)}=\dfrac{\displaystyle\lim_{x\to{a}}f(x)}{\displaystyle\lim_{x\to{a}}g(x)}$$, if $$\displaystyle\lim_{x\to{a}}g(x)\neq 0$$

Limit of an exponential function $$\displaystyle\lim_{x\to{a}}{a^{f\left(x\right)}}=a^{\displaystyle\lim_{x\to{a}}{f\left(x\right)}}$$

Common limits $$\displaystyle\lim_{x\to{0}}{\dfrac{\sin\left(x\right)}{x}}=1$$ and $$\displaystyle\lim_{x\to{\infty}}{(1+\dfrac{1}{x})^x}=e$$

The squeeze theorem: if $$g\left(x\right)\leq f\left(x\right)\leq h\left(x\right)$$ and $$\displaystyle\lim_{x\to{a}}g(x)=\lim_{x\to{a}}h(x)=L\;\Rightarrow\;\lim_{x\to{a}}f(x)=L$$

L'Hôpital's rule: if $$\displaystyle\lim_{x\to{a}}f(x)=0$$ and $$\displaystyle\lim_{x\to{a}}g(x)=0$$ (or both equal to $$\infty$$), then $$\displaystyle\lim_{x\to{a}}{\dfrac{f\left(x\right)}{g\left(x\right)}}=\lim_{x\to{a}}{\dfrac{f'\left(x\right)}{g'\left(x\right)}}$$

Taylor series $$f(x)=\sum\limits_{n=0}^{\infty}\dfrac{f^{\left(n\right)}\left(a\right)}{n!}\,\left(x-a\right)^n$$

Applies multiplication by conjugate, substitutions and Euler's formula

Evaluates both two-sided $$x\to{a}$$ and one-sided $$x\to{a+}$$ limits

## Complex numbers

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Calculator converts a complex expression $$f(z)$$ to its algebraic $$z=a+i\,b$$, trigonometric $$z=r\cdot(\cos(\varphi)+i\,\sin(\varphi))$$ and exponential form $$z=r\,e^{i\,\varphi}$$ using:

Modulus of a complex number $$r=\left|a+i\,b\right|=\sqrt{a^2+b^2}$$

Root of a complex number $$\sqrt[n]{z}=\sqrt[n]{r}\,\left(\cos\left(\dfrac{\varphi+2\,\pi\,\mathrm{k}}{n}\right)+i\,\sin\left(\dfrac{\varphi+2\,\pi\,\mathrm{k}}{n}\right)\right)$$

Power of a complex number $$z^n=r^n\,\left(\cos\left(n\,\varphi\right)+i\,\sin\left(n\,\varphi\right)\right)$$

Multiplying a fraction by its conjugate $$\dfrac{z}{a+i\,b}\;\Rightarrow\;\dfrac{z\cdot\left(a-i\,b\right)}{\left(a+i\,b\right)\cdot\left(a-i\,b\right)}\;\Rightarrow\;\dfrac{z\cdot\left(a-i\,b\right)}{a^2+b^2}$$

Complex logarithm $$\operatorname{Log}\left(z\right)=\ln\left(r\right)+i\,(\varphi+2\,\pi\,\mathrm{k})$$

The principal value of the complex logarithm $$\mathrm{Im}\operatorname{Log}\in(-\pi,\,\pi]$$

Trigonometric and hyperbolic formulas like $$\sin\left(\alpha\pm\beta\right)=\sin\left(\alpha\right)\,\cos\left(\beta\right)\pm\cos\left(\alpha\right)\,\sin\left(\beta\right)$$ or $$\sinh\left(i\,b\right)=i\,\sin\left(b\right)$$, and Euler's formula $$e^{i\,\varphi}=\cos\left(\varphi\right)+i\,\sin\left(\varphi\right)$$

## Matrix calculations

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Calculator calculates given matrix expressions with matrices $$\mathrm{A}$$, $$\mathrm{B}$$ and $$\mathrm{C}$$

Its functionality includes such matrix operations as: addition $$\mathrm{A}+\mathrm{B}$$, subtraction $$\mathrm{A}-\mathrm{B}$$, multiplication $$\mathrm{C}\cdot\mathrm{B}$$, determinant $$\left|\mathrm{A}\right|$$, transpose $$\mathrm{B}^{\mathrm{T}}$$, rank $$\operatorname{rank}\mathrm{C}$$, inversion $$\mathrm{A}^{-1}$$, multiplication by a constant $$a\cdot\mathrm{B}$$ or addition with a constant $$c+\mathrm{A}$$

Calculates the derivative of matrix elements $$\left(\mathrm{C}\right)'_x={\scriptsize\left(\begin{gathered}\left(\mathrm{a_{11}}\right)'_x&\left(\mathrm{a_{12}}\right)'_x\\\left(\mathrm{a_{21}}\right)'_x&\left(\mathrm{a_{22}}\right)'_x\end{gathered}\right)}$$ or integral of matrix elements $$\int{\mathrm{A}}{\;\mathrm{d}x}={\scriptsize\left(\begin{gathered}\int{\mathrm{a_{11}}}{\;\mathrm{d}x}&\int{\mathrm{a_{12}}}{\;\mathrm{d}x}\\\int{\mathrm{a_{21}}}{\;\mathrm{d}x}&\int{\mathrm{a_{22}}}{\;\mathrm{d}x}\end{gathered}\right)}$$

Applies the mathematical functions $$\sin$$, $$\cos$$$$\,\ldots$$ to a matrix element by element, for example $$\ln\left(\mathrm{A}\right)={\scriptsize\left(\begin{gathered}\ln\left(\mathrm{a_{11}}\right)&\ln\left(\mathrm{a_{12}}\right)\\\ln\left(\mathrm{a_{21}}\right)&\ln\left(\mathrm{a_{22}}\right)\end{gathered}\right)}$$

Evaluates both numeric values and combinations of arithmetic operations and functions

## Numerical calculator

π
ln
sin
sinh
e
log2
cos
cosh
φ
log
tan
tanh
°
|x|
cot
coth
ex
sin⁻¹
sinh⁻¹
²
2x
cos⁻¹
cosh⁻¹
³
10x
tan⁻¹
tanh⁻¹
x!
cot⁻¹
coth⁻¹
C
7
4
1
,
( )
8
5
2
0
%
9
6
3
÷
×
+
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