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- Linear homogeneous differential equations of 2nd order Step-By-Step
- Linear inhomogeneous differential equations of the 1st order Step-By-Step
- Differential equations with separable variables Step-by-Step
- A simplest differential equations of 1-order Step-by-Step
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How to use it?
- Differential equation:
- Equation xy^3dx+e^(x^2)dy=0
- Equation xydx-(x+2y)^2dy=0
- Equation (1+4xy+2y^2)dx+(1+4xy+2x^2)dy=0
- Equation 2x(y+1)dx-ydy=0
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Identical expressions
- xy^3dx+e^(x^ two)dy= zero
- xy cubed dx plus e to the power of (x squared )dy equally 0
- xy cubed dx plus e to the power of (x to the power of two)dy equally zero
- xy3dx+e(x2)dy=0
- xy3dx+ex2dy=0
- xy³dx+e^(x²)dy=0
- xy to the power of 3dx+e to the power of (x to the power of 2)dy=0
- xy^3dx+e^x^2dy=0
- xy^3dx+e^(x^2)dy=O
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Similar expressions
- xy^3dx-e^(x^2)dy=0
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Expressions with functions
- xy
- xydx-(x+2y)^2dy=0
- xy'+4y=8x^4
- xydx-(x^2-3y^2)dy=0
- xy'=y*(ln(y)-ln(x))
- xy''-(2x+1)y'+(x+1)y=0
- dy
- dy/dx=(x+3y)/(3x+y)
- dy/dx=xy^(1/2)
- dy/dx=4y-8
- dy/dx=sin(y)
- dy/dx=y^(2/3)
Differential equation xy^3dx+e^(x^2)dy=0
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The solution
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[src]
/ 2\
3 d \x /
x*y (x) + --(y(x))*e = 0
dx
$$x y^{3}{\left(x \right)} + e^{x^{2}} \frac{d}{d x} y{\left(x \right)} = 0$$
x*y^3 + exp(x^2)*y' = 0
Detail solution
Given the equation:
$$x y^{3}{\left(x \right)} + e^{x^{2}} \frac{d}{d x} y{\left(x \right)} = 0$$
This differential equation has the form:
where
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = - x e^{- x^{2}}$$
$$\operatorname{g_{2}}{\left(y \right)} = y^{3}{\left(x \right)}$$
We give the equation to the form:
Divide both parts of the equation by g2(y)
$$y^{3}{\left(x \right)}$$
we get
$$\frac{\frac{d}{d x} y{\left(x \right)}}{y^{3}{\left(x \right)}} = - x e^{- x^{2}}$$
We separated the variables x and y.
Now, multiply the both equation sides by dx,
then the equation will be the
$$\frac{dx \frac{d}{d x} y{\left(x \right)}}{y^{3}{\left(x \right)}} = - dx x e^{- x^{2}}$$
or
$$\frac{dy}{y^{3}{\left(x \right)}} = - dx x e^{- x^{2}}$$
Take the integrals from the both equation sides:
- the integral of the left side by y,
- the integral of the right side by x.
$$\int \frac{1}{y^{3}}\, dy = \int \left(- x e^{- x^{2}}\right)\, dx$$
Detailed solution of the integral with y
Detailed solution of the integral with x
Take this integrals
$$- \frac{1}{2 y^{2}} = Const + \frac{e^{- x^{2}}}{2}$$
Detailed solution of the equation
We get the simple equation with the unknown variable y.
(Const - it is a constant)
Solution is:
$$\operatorname{y_{1}} = y{\left(x \right)} = - \sqrt{- \frac{e^{x^{2}}}{C_{1} e^{x^{2}} + 1}}$$
$$\operatorname{y_{2}} = y{\left(x \right)} = \sqrt{- \frac{e^{x^{2}}}{C_{1} e^{x^{2}} + 1}}$$
$$x y^{3}{\left(x \right)} + e^{x^{2}} \frac{d}{d x} y{\left(x \right)} = 0$$
This differential equation has the form:
f1(x)*g1(y)*y' = f2(x)*g2(y),
where
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = - x e^{- x^{2}}$$
$$\operatorname{g_{2}}{\left(y \right)} = y^{3}{\left(x \right)}$$
We give the equation to the form:
g1(y)/g2(y)*y'= f2(x)/f1(x).
Divide both parts of the equation by g2(y)
$$y^{3}{\left(x \right)}$$
we get
$$\frac{\frac{d}{d x} y{\left(x \right)}}{y^{3}{\left(x \right)}} = - x e^{- x^{2}}$$
We separated the variables x and y.
Now, multiply the both equation sides by dx,
then the equation will be the
$$\frac{dx \frac{d}{d x} y{\left(x \right)}}{y^{3}{\left(x \right)}} = - dx x e^{- x^{2}}$$
or
$$\frac{dy}{y^{3}{\left(x \right)}} = - dx x e^{- x^{2}}$$
Take the integrals from the both equation sides:
- the integral of the left side by y,
- the integral of the right side by x.
$$\int \frac{1}{y^{3}}\, dy = \int \left(- x e^{- x^{2}}\right)\, dx$$
Detailed solution of the integral with y
Detailed solution of the integral with x
Take this integrals
$$- \frac{1}{2 y^{2}} = Const + \frac{e^{- x^{2}}}{2}$$
Detailed solution of the equation
We get the simple equation with the unknown variable y.
(Const - it is a constant)
Solution is:
$$\operatorname{y_{1}} = y{\left(x \right)} = - \sqrt{- \frac{e^{x^{2}}}{C_{1} e^{x^{2}} + 1}}$$
$$\operatorname{y_{2}} = y{\left(x \right)} = \sqrt{- \frac{e^{x^{2}}}{C_{1} e^{x^{2}} + 1}}$$
The answer
[src]
______________
/ / 2\
/ \x /
/ -e
y(x) = - / ------------
/ / 2\
/ \x /
\/ 1 + C1*e
$$y{\left(x \right)} = - \sqrt{- \frac{e^{x^{2}}}{C_{1} e^{x^{2}} + 1}}$$
______________
/ / 2\
/ \x /
/ -e
y(x) = / ------------
/ / 2\
/ \x /
\/ 1 + C1*e
$$y{\left(x \right)} = \sqrt{- \frac{e^{x^{2}}}{C_{1} e^{x^{2}} + 1}}$$
The classification
separable
1st power series
lie group
separable Integral
The graph