Here, we show you a step-by-step solved example of matrices. This solution was automatically generated by our smart calculator:
Rewrite the fraction $\frac{1}{x\left(x-1\right)}$ in $2$ simpler fractions using partial fraction decomposition
Find the values for the unknown coefficients: $A, B$. The first step is to multiply both sides of the equation from the previous step by $x\left(x-1\right)$
Multiplying polynomials
Simplifying
Assigning values to $x$ we obtain the following system of equations
Proceed to solve the system of linear equations
Rewrite as a coefficient matrix
Reducing the original matrix to a identity matrix using Gaussian Elimination
The integral of $\frac{1}{x\left(x-1\right)}$ in decomposed fractions equals
Expand the integral $\int\left(\frac{-1}{x}+\frac{1}{x-1}\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
Solve the integral $\int\frac{1}{y}dy$ and replace the result in the differential equation
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
Apply the formula: $\int\frac{n}{x+b}dx$$=nsign\left(x\right)\ln\left(x+b\right)+C$, where $b=-1$ and $n=1$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Solve the integral $\int\frac{-1}{x}dx+\int\frac{1}{x-1}dx$ and replace the result in the differential equation
The difference of two logarithms of equal base $b$ is equal to the logarithm of the quotient: $\log_b(x)-\log_b(y)=\log_b\left(\frac{x}{y}\right)$
Take the variable outside of the logarithm
Simplifying the logarithm
Simplify $e^{\left(\ln\left(\frac{x-1}{x}\right)+C_0\right)}$ by applying the properties of exponents and logarithms
Multiplying the fraction by $e^{C_0}$
We can rename $e^{C_0}$ as other constant
Find the explicit solution to the differential equation. We need to isolate the variable $y$
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