Step-by-step calculators:
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
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:
\(\ln\) \(\sin\) \(\cos\) \(\tan\) \(\cot\) \(\arctan\) \(\arcsin\) \(\arccos\) \(\operatorname{arccot}\) \(\sinh\) \(\cosh\) \(\tanh\) \(\coth\) \(\operatorname{sech}\) \(\operatorname{csch}\) \(\operatorname{arsinh}\) \(\operatorname{arcosh}\) \(\operatorname{artanh}\) \(\operatorname{arcoth}\) \(\operatorname{arcsec}\) \(\operatorname{arccsc}\) \(\operatorname{arsech}\) \(\operatorname{arcsch}\) \(\sec\) \(\csc\) \(\left|f\right|\)
Collection of solved indefinite integrals: Google Drive .pdf
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 \(\operatorname{sgn}\left(f\right)\)
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