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How to use it?
- Differential equation:
- Equation dx+xydy=(y^2)dx+ydy
- Equation (2xy^4+sin(y))dx+(4x^2y^3+xcos(y))dy=0
- Equation y''+2y'+y=2e^(-x)
- Equation y'-y=5.2
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Identical expressions
- dx+xydy=(y^ two)dx+ydy
- dx plus xydy equally (y squared )dx plus ydy
- dx plus xydy equally (y to the power of two)dx plus ydy
- dx+xydy=(y2)dx+ydy
- dx+xydy=y2dx+ydy
- dx+xydy=(y²)dx+ydy
- dx+xydy=(y to the power of 2)dx+ydy
- dx+xydy=y^2dx+ydy
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Similar expressions
- dx-xydy=(y^2)dx+ydy
- dx+xydy=(y^2)dx-ydy
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Expressions with functions
- dx
- dx/dt=r-kx
- dx/dt+x/t^2=1/t^2
- dx/dt=3xt^2
- dx/dt+3x/20+t=4
- dx/dt=-k(a-x)
- xydy
- xydy=y^2dx+x^2dy
- xydy=(y+1)(1-x)dx
- xydy+dx=y²dx+ydy
- xydy-(x+1)sqrt(1-y^2)dx=0
- xydy-(x2+y2)dx=0
- dx
- dx/dt=r-kx
- dx/dt+x/t^2=1/t^2
- dx/dt=3xt^2
- dx/dt+3x/20+t=4
- dx/dt=-k(a-x)
- ydy
- ydy-xdx=0
- ydy=dx
- ydy-4xdx=0
- ydy=2xdx
- ydy-xdx=dx
Differential equation dx+xydy=(y^2)dx+ydy
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The solution
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d 2 d
1 + x*--(y(x))*y(x) = y (x) + --(y(x))*y(x)
dx dx
$$x y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + 1 = y^{2}{\left(x \right)} + y{\left(x \right)} \frac{d}{d x} y{\left(x \right)}$$
x*y*y' + 1 = y^2 + y*y'
Detail solution
Given the equation:
$$x y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} - y^{2}{\left(x \right)} - y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + 1 = 0$$
This differential equation has the form:
where
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = - \frac{1}{x - 1}$$
$$\operatorname{g_{2}}{\left(y \right)} = - \frac{y^{2}{\left(x \right)} - 1}{y{\left(x \right)}}$$
We give the equation to the form:
Divide both parts of the equation by g2(y)
$$- \frac{y^{2}{\left(x \right)} - 1}{y{\left(x \right)}}$$
we get
$$- \frac{y{\left(x \right)} \frac{d}{d x} y{\left(x \right)}}{y^{2}{\left(x \right)} - 1} = - \frac{1}{x - 1}$$
We separated the variables x and y.
Now, multiply the both equation sides by dx,
then the equation will be the
$$- \frac{dx y{\left(x \right)} \frac{d}{d x} y{\left(x \right)}}{y^{2}{\left(x \right)} - 1} = - \frac{dx}{x - 1}$$
or
$$- \frac{dy y{\left(x \right)}}{y^{2}{\left(x \right)} - 1} = - \frac{dx}{x - 1}$$
Take the integrals from the both equation sides:
- the integral of the left side by y,
- the integral of the right side by x.
$$\int \left(- \frac{y}{y^{2} - 1}\right)\, dy = \int \left(- \frac{1}{x - 1}\right)\, dx$$
Detailed solution of the integral with y
Detailed solution of the integral with x
Take this integrals
$$- \frac{\log{\left(y^{2} - 1 \right)}}{2} = Const - \log{\left(x - 1 \right)}$$
Detailed solution of the equation
We get the simple equation with the unknown variable y.
(Const - it is a constant)
Solution is:
$$\operatorname{y_{1}} = y{\left(x \right)} = - \sqrt{C_{1} x^{2} - 2 C_{1} x + C_{1} + 1}$$
$$\operatorname{y_{2}} = y{\left(x \right)} = \sqrt{C_{1} x^{2} - 2 C_{1} x + C_{1} + 1}$$
$$x y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} - y^{2}{\left(x \right)} - y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + 1 = 0$$
This differential equation has the form:
f1(x)*g1(y)*y' = f2(x)*g2(y),
where
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = - \frac{1}{x - 1}$$
$$\operatorname{g_{2}}{\left(y \right)} = - \frac{y^{2}{\left(x \right)} - 1}{y{\left(x \right)}}$$
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)
$$- \frac{y^{2}{\left(x \right)} - 1}{y{\left(x \right)}}$$
we get
$$- \frac{y{\left(x \right)} \frac{d}{d x} y{\left(x \right)}}{y^{2}{\left(x \right)} - 1} = - \frac{1}{x - 1}$$
We separated the variables x and y.
Now, multiply the both equation sides by dx,
then the equation will be the
$$- \frac{dx y{\left(x \right)} \frac{d}{d x} y{\left(x \right)}}{y^{2}{\left(x \right)} - 1} = - \frac{dx}{x - 1}$$
or
$$- \frac{dy y{\left(x \right)}}{y^{2}{\left(x \right)} - 1} = - \frac{dx}{x - 1}$$
Take the integrals from the both equation sides:
- the integral of the left side by y,
- the integral of the right side by x.
$$\int \left(- \frac{y}{y^{2} - 1}\right)\, dy = \int \left(- \frac{1}{x - 1}\right)\, dx$$
Detailed solution of the integral with y
Detailed solution of the integral with x
Take this integrals
$$- \frac{\log{\left(y^{2} - 1 \right)}}{2} = Const - \log{\left(x - 1 \right)}$$
Detailed solution of the equation
We get the simple equation with the unknown variable y.
(Const - it is a constant)
Solution is:
$$\operatorname{y_{1}} = y{\left(x \right)} = - \sqrt{C_{1} x^{2} - 2 C_{1} x + C_{1} + 1}$$
$$\operatorname{y_{2}} = y{\left(x \right)} = \sqrt{C_{1} x^{2} - 2 C_{1} x + C_{1} + 1}$$
The answer
[src]
_________________________
/ 2
y(x) = -\/ 1 + C1 + C1*x - 2*C1*x
$$y{\left(x \right)} = - \sqrt{C_{1} x^{2} - 2 C_{1} x + C_{1} + 1}$$
_________________________
/ 2
y(x) = \/ 1 + C1 + C1*x - 2*C1*x
$$y{\left(x \right)} = \sqrt{C_{1} x^{2} - 2 C_{1} x + C_{1} + 1}$$
Graph of the Cauchy problem
The classification
separable
1st exact
1st power series
lie group
separable Integral
1st exact Integral
Numerical answer
[src]
(x, y):
(-10.0, 0.75)
(-7.777777777777778, 0.8493599517053494)
(-5.555555555555555, 0.919028857717191)
(-3.333333333333333, 0.965455921734598)
(-1.1111111111111107, 0.9919100594412996)
(1.1111111111111107, 0.9999777939747626)
(3.333333333333334, 0.990169095120081)
(5.555555555555557, 0.9619883863793672)
(7.777777777777779, 0.9137373526062414)
(10.0, 0.8419725920923582)
(10.0, 0.8419725920923582)
The graph