Mister Exam

# Differential equation dy/dx=y^(2/3)

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

from to

### The solution

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d           2/3
--(y(x)) = y   (x)
dx                
$$\frac{d}{d x} y{\left(x \right)} = y^{\frac{2}{3}}{\left(x \right)}$$
y' = y^(2/3)
Detail solution
Given the equation:
$$\frac{d}{d x} y{\left(x \right)} = y^{\frac{2}{3}}{\left(x \right)}$$
This differential equation has the form:
$$f_1(x)*g_1(y)*y' = f_2(x)*g_2(y)$$
where
$$f_{1}{\left(x \right)} = 1$$
$$g_{1}{\left(y \right)} = 1$$
$$f_{2}{\left(x \right)} = 1$$
$$g_{2}{\left(y \right)} = y^{\frac{2}{3}}{\left(x \right)}$$
We give the equation to the form:
$$\frac{g_1(y)}{g_2(y)}*y'= \frac{f_2(x)}{f_1(x)}$$
Divide both parts of the equation by $g_{2}{\left(y{\left(x \right)} \right)}$
$$y^{\frac{2}{3}}{\left(x \right)}$$
we get
$$\frac{\frac{d}{d x} y{\left(x \right)}}{y^{\frac{2}{3}}{\left(x \right)}} = 1$$
We separated the variables x and y.

Now, multiply the both equation sides by dx,
then the equation will be the
$$\frac{dx \frac{d}{d x} y{\left(x \right)}}{y^{\frac{2}{3}}{\left(x \right)}} = dx$$
or
$$\frac{dy}{y^{\frac{2}{3}}{\left(x \right)}} = dx$$

Take the integrals from the both equation sides:
- the integral of the left side by y,
- the integral of the right side by x.
$$\int \frac{1}{y^{\frac{2}{3}}}\, dy = \int 1\, dx$$
Detailed solution of the integral with y
Detailed solution of the integral with x
Take this integrals
$$3 \sqrt[3]{y} = Const + x$$
Detailed solution of the equation
We get the simple equation with the unknown variable y.
(Const - it is a constant)

Solution is:
$$y_{1} = y{\left(x \right)} = \frac{C_{1}^{3}}{27} + \frac{C_{1}^{2} x}{9} + \frac{C_{1} x^{2}}{9} + \frac{x^{3}}{27}$$
$$y\left(x\right)={\it ilt}\left({{\mathcal{L}\left(y\left(x\right)^{ {{2}\over{3}}} , x , g_{19164}\right)+y\left(0\right)}\over{ g_{19164}}} , g_{19164} , x\right)$$
y = 'ilt(('laplace(y^(2/3),x,g19164)+y(0))/g19164,g19164,x)
         3    3       2       2
C1    x    C1*x    x*C1
y(x) = --- + -- + ----- + -----
27   27     9       9  
$$y{\left(x \right)} = \frac{C_{1}^{3}}{27} + \frac{C_{1}^{2} x}{9} + \frac{C_{1} x^{2}}{9} + \frac{x^{3}}{27}$$
Graph of the Cauchy problem
The classification
1st exact
1st exact Integral
1st power series
Bernoulli
Bernoulli Integral
lie group
separable
separable Integral
(x, y):
(-10.0, 0.75)
(-7.777777777777778, 4.486418845694451)
(-5.555555555555555, 13.652635365245867)
(-3.333333333333333, 30.687302265806366)
(-1.1111111111111107, 58.02907219178992)
(1.1111111111111107, 98.11659778761066)
(3.333333333333334, 153.38853169768208)
(5.555555555555557, 226.2835265664187)
(7.777777777777779, 319.24023503823406)
(10.0, 434.69730975754226)
(10.0, 434.69730975754226)
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