Step-by-step calculators:

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)

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:

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

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 \(f\left(g\left(x\right)\right) = f\left(r\left(x\right)\right)\;\Rightarrow\;g\left(x\right)=r\left(x\right)\)

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

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

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

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 \(e^{i\,\varphi}=\cos\left(\varphi\right)+i\,\sin\left(\varphi\right)\)

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