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xy^3dx+e^(x^2)dy=0

Differential equation xy^3dx+e^(x^2)dy=0

For Cauchy problem:

y() =
y'() =
y''() =
y'''() =
y''''() =

The graph:

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The solution

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                    / 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:
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
Differential equation xy^3dx+e^(x^2)dy=0
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