Trennbare Differentialgleichungen Rechner

Mit unserem Trennbare Differentialgleichungen Schritt-für-Schritt-Rechner erhalten Sie detaillierte Lösungen für Ihre mathematischen Probleme. Üben Sie Ihre mathematischen Fähigkeiten und lernen Sie Schritt für Schritt mit unserem Mathe-Löser. Alle unsere Online-Rechner finden Sie hier.

Geben Sie ein mathematisches Problem oder eine Frage ein



Symbolischer Modus

Text-Modus

Go!



(◻)

◻/◻

◻²

◻^◻

√◻

√

^◻√◻

^◻√

∞

log

log_◻

lim

d/dx

D^□_x

∫

∫^◻

|◻|

sin

cos

tan

cot

sec

csc

asin

acos

atan

acot

asec

acsc

sinh

cosh

tanh

coth

sech

csch

asinh

acosh

atanh

acoth

asech

acsch

 Schwierige Probleme

Here, we show you a step-by-step solved example of separable differential equations. This solution was automatically generated by our smart calculator:

\left(2xy-y\right)dx+\left(x^2+x\right)dy=0

Factoring by $y$

y\left(2x-1\right)dx+\left(x^2+x\right)dy=0

Grouping the terms of the differential equation

\left(x^2+x\right)dy=-\left(2x-1\right)y\cdot dx

Group the terms of the differential equation. Move the terms of the $y$ variable to the left side, and the terms of the $x$ variable to the right side of the equality

\frac{1}{y}dy=\frac{-\left(2x-1\right)}{x^2+x}dx

Factor the polynomial $x^2+x$ by it's greatest common factor (GCF): $x$

\frac{-\left(2x-1\right)}{x\left(x+1\right)}dx

Simplify the expression $\frac{-\left(2x-1\right)}{x^2+x}dx$

\frac{1}{y}dy=\frac{-\left(2x-1\right)}{x\left(x+1\right)}dx

Integrate both sides of the differential equation, the left side with respect to $y$ , and the right side with respect to $x$

\int\frac{1}{y}dy=\int\frac{-\left(2x-1\right)}{x\left(x+1\right)}dx

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

\ln\left|y\right|

Solve the integral $\int\frac{1}{y}dy$ and replace the result in the differential equation

\ln\left|y\right|=\int\frac{-\left(2x-1\right)}{x\left(x+1\right)}dx

Take out the constant $-1$ from the integral

-\int\frac{2x-1}{x\left(x+1\right)}dx

Rewrite the fraction $\frac{2x-1}{x\left(x+1\right)}$ in $2$ simpler fractions using partial fraction decomposition

\frac{-1}{x}+\frac{3}{x+1}

Expand the integral $\int\left(\frac{-1}{x}+\frac{3}{x+1}\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

-\int\frac{-1}{x}dx-\int\frac{3}{x+1}dx

We can solve the integral $\int\frac{3}{x+1}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$ ), which when substituted makes the integral easier. We see that $x+1$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

u=x+1

Now, in order to rewrite $dx$ in terms of $du$ , we need to find the derivative of $u$ . We need to calculate $du$ , we can do that by finding the derivative of the equation above

du=dx

Substituting $u$ and $dx$ in the integral and simplify

-\int\frac{-1}{x}dx-\int\frac{3}{u}du

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

- -\ln\left|x\right|-\int\frac{3}{u}du

Multiply $-1$ times $-1$

\ln\left|x\right|-\int\frac{3}{u}du

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

\ln\left|x\right|- 3\ln\left|u\right|

Multiply $-1$ times $3$

\ln\left|x\right|-3\ln\left|u\right|

Replace $u$ with the value that we assigned to it in the beginning: $x+1$

\ln\left|x\right|-3\ln\left|x+1\right|

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

\ln\left|x\right|-3\ln\left|x+1\right|+C_0

Solve the integral $\int\frac{-\left(2x-1\right)}{x\left(x+1\right)}dx$ and replace the result in the differential equation

\ln\left|y\right|=\ln\left|x\right|-3\ln\left|x+1\right|+C_0

Take the variable outside of the logarithm

e^{\ln\left(y\right)}=e^{\left(\ln\left(x\right)-3\ln\left(x+1\right)+C_0\right)}

Simplifying the logarithm

y=e^{\left(\ln\left(x\right)-3\ln\left(x+1\right)+C_0\right)}

Simplify $e^{\left(\ln\left(x\right)-3\ln\left(x+1\right)+C_0\right)}$ by applying the properties of exponents and logarithms

y=xe^{\left(-3\ln\left(x+1\right)+C_0\right)}

Simplify $e^{\left(-3\ln\left(x+1\right)+C_0\right)}$ by applying the properties of exponents and logarithms

y=e^{C_0}x\left(x+1\right)^{-3}

We can rename $e^{C_0}$ as other constant

y=C_1x\left(x+1\right)^{-3}

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$ , where $n$ is a number

y=C_1x\frac{1}{\left(x+1\right)^{3}}

Multiply the fraction by the term

y=\frac{C_1x}{\left(x+1\right)^{3}}

Find the explicit solution to the differential equation. We need to isolate the variable $y$

y=\frac{C_1x}{\left(x+1\right)^{3}}

 Endgültige Antwort auf das Problem

y=\frac{C_1x}{\left(x+1\right)^{3}}

Trennbare Differentialgleichungen Rechner

Mit unserem Trennbare Differentialgleichungen Schritt-für-Schritt-Rechner erhalten Sie detaillierte Lösungen für Ihre mathematischen Probleme. Üben Sie Ihre mathematischen Fähigkeiten und lernen Sie Schritt für Schritt mit unserem Mathe-Löser. Alle unsere Online-Rechner finden Sie hier.

 Beispiel

 Gelöste Probleme

 Schwierige Probleme

 Endgültige Antwort auf das Problem

Haben Sie Probleme mit Mathematik?

 Beliebte Probleme

Trennbare Differentialgleichungen Rechner

Mit unserem Trennbare Differentialgleichungen Schritt-für-Schritt-Rechner erhalten Sie detaillierte Lösungen für Ihre mathematischen Probleme. Üben Sie Ihre mathematischen Fähigkeiten und lernen Sie Schritt für Schritt mit unserem Mathe-Löser. Alle unsere Online-Rechner finden Sie hier.

 Beispiel

 Gelöste Probleme

 Schwierige Probleme

 Endgültige Antwort auf das Problem

Haben Sie Probleme mit Mathematik?

 Verwandte Rechner

 Beliebte Probleme