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Other calculators
- 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 dy/dx=y^(2/3)
- Equation y"+5y'-6y=0
- Equation 4y"+12y'+9y=0
- Equation y'''+3y''-4y'-12y=0
- Derivative of:
-
dy/dx
- Integral of d{x}:
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y^(2/3)
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Identical expressions
- dy/dx=y^(two / three)
- dy divide by dx equally y to the power of (2 divide by 3)
- dy divide by dx equally y to the power of (two divide by three)
- dy/dx=y(2/3)
- dy/dx=y2/3
- dy/dx=y^2/3
- dy divide by dx=y^(2 divide by 3)
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Similar expressions
- y'=(x^2+3xy-y^2)/(3x^2-2xy)
- sinxdx*y^2/3*dy=0
- xy'=(2y^3+4y^2)/(3y^2+6x^2)
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Expressions with functions
- dy
- dy/dx=(x^2+3y^2)/(2xy)
- dy/dx-y=e^xy^2
- dy/dx=4y-8
- dy/dx=5y
- dy/dx=ycosx/1+2y^2
- dx
- dx/dt=x
- dx/dt=3xt^2
- dx/dt=-k(a-x)
- dx/dt+3x/20+t=4
- dx/dt=kx(n+1-x)
Differential equation dy/dx=y^(2/3)
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The solution
You have entered
[src]
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:
$$y_{1} = y{\left(x \right)} = \frac{C_{1}^{3}}{27} + \frac{C_{1}^{2} x}{9} + \frac{C_{1} x^{2}}{9} + \frac{x^{3}}{27}$$
$$\frac{d}{d x} y{\left(x \right)} = y^{\frac{2}{3}}{\left(x \right)}$$
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:
$$y_{1} = y{\left(x \right)} = \frac{C_{1}^{3}}{27} + \frac{C_{1}^{2} x}{9} + \frac{C_{1} x^{2}}{9} + \frac{x^{3}}{27}$$
The answer (#2)
[src]
$$y\left(x\right)={\it ilt}\left({{\mathcal{L}\left(y\left(x\right)^{
{{2}\over{3}}} , x , g_{19164}\right)+y\left(0\right)}\over{
g_{19164}}} , g_{19164} , x\right)$$
y = 'ilt(('laplace(y^(2/3),x,g19164)+y(0))/g19164,g19164,x)
The answer
[src]
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
Numerical answer
[src]
(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)