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- Differential equations with separable variables Step-by-Step
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How to use it?
- Differential equation:
- Equation dx/dt=ax-bx^2
- Equation dy/dx=(x+3y)/(3x+y)
- Equation dx+xydy=(y^2)dx+ydy
- Equation (2xy^4+sin(y))dx+(4x^2y^3+xcos(y))dy=0
- Derivative of:
- dx/dt
-
Identical expressions
- dx/dt=ax-bx^ two
- dx divide by dt equally ax minus bx squared
- dx divide by dt equally ax minus bx to the power of two
- dx/dt=ax-bx2
- dx/dt=ax-bx²
- dx/dt=ax-bx to the power of 2
- dx divide by dt=ax-bx^2
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Similar expressions
- dx/dt=ax+bx^2
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Expressions with functions
- dx
- dx+xydy=(y^2)dx+ydy
- dx/dt=r-kx
- dx/dt+x/t^2=1/t^2
- dx/dt=3xt^2
- dx/dt+3x/20+t=4
- dt
- dt*t*x/dx=t^2-1
- dt=5x-3
- dt=dy+csc(y)dt
- dt*(3*t+y4)+4*dy*t*y^3=0
- dt*(e^t+t^2*y)*y-dy*e^t=0
- ax
- axy'+by'=0
- ax"+by'+cy=0
- axy''+2ay'=-bx^2
- ax^2+b(x')^2=c
- axy(y')^2+(x^2-ay^2-b)y'=xy
Differential equation dx/dt=ax-bx^2
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The solution
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d 2 --(x(t)) = a*x(t) - b*x (t) dt
$$\frac{d}{d t} x{\left(t \right)} = a x{\left(t \right)} - b x^{2}{\left(t \right)}$$
x' = a*x - b*x^2
Detail solution
Given the equation:
$$\frac{d}{d t} x{\left(t \right)} = a x{\left(t \right)} - b x^{2}{\left(t \right)}$$
This differential equation has the form:
where
$$\operatorname{f_{1}}{\left(t \right)} = 1$$
$$\operatorname{g_{1}}{\left(x \right)} = 1$$
$$\operatorname{f_{2}}{\left(t \right)} = 1$$
$$\operatorname{g_{2}}{\left(x \right)} = \left(a - b x{\left(t \right)}\right) x{\left(t \right)}$$
We give the equation to the form:
Divide both parts of the equation by g2(x)
$$\left(a - b x{\left(t \right)}\right) x{\left(t \right)}$$
we get
$$\frac{\frac{d}{d t} x{\left(t \right)}}{\left(a - b x{\left(t \right)}\right) x{\left(t \right)}} = 1$$
We separated the variables t and x.
Now, multiply the both equation sides by dt,
then the equation will be the
$$\frac{dt \frac{d}{d t} x{\left(t \right)}}{\left(a - b x{\left(t \right)}\right) x{\left(t \right)}} = dt$$
or
$$\frac{dx}{\left(a - b x{\left(t \right)}\right) x{\left(t \right)}} = dt$$
Take the integrals from the both equation sides:
- the integral of the left side by x,
- the integral of the right side by t.
$$\int \frac{1}{x \left(a - b x\right)}\, dx = \int 1\, dt$$
Detailed solution of the integral with x
Detailed solution of the integral with t
Take this integrals
$$- \frac{- \log{\left(x \right)} + \log{\left(- \frac{a}{b} + x \right)}}{a} = Const + t$$
Detailed solution of the equation
We get the simple equation with the unknown variable x.
(Const - it is a constant)
Solution is:
$$\operatorname{x_{1}} = x{\left(t \right)} = \frac{a e^{a \left(C_{1} + t\right)}}{b \left(e^{a \left(C_{1} + t\right)} - 1\right)}$$
$$\frac{d}{d t} x{\left(t \right)} = a x{\left(t \right)} - b x^{2}{\left(t \right)}$$
This differential equation has the form:
f1(x)*g1(x)*x' = f2(x)*g2(x),
where
$$\operatorname{f_{1}}{\left(t \right)} = 1$$
$$\operatorname{g_{1}}{\left(x \right)} = 1$$
$$\operatorname{f_{2}}{\left(t \right)} = 1$$
$$\operatorname{g_{2}}{\left(x \right)} = \left(a - b x{\left(t \right)}\right) x{\left(t \right)}$$
We give the equation to the form:
g1(x)/g2(x)*x'= f2(x)/f1(x).
Divide both parts of the equation by g2(x)
$$\left(a - b x{\left(t \right)}\right) x{\left(t \right)}$$
we get
$$\frac{\frac{d}{d t} x{\left(t \right)}}{\left(a - b x{\left(t \right)}\right) x{\left(t \right)}} = 1$$
We separated the variables t and x.
Now, multiply the both equation sides by dt,
then the equation will be the
$$\frac{dt \frac{d}{d t} x{\left(t \right)}}{\left(a - b x{\left(t \right)}\right) x{\left(t \right)}} = dt$$
or
$$\frac{dx}{\left(a - b x{\left(t \right)}\right) x{\left(t \right)}} = dt$$
Take the integrals from the both equation sides:
- the integral of the left side by x,
- the integral of the right side by t.
$$\int \frac{1}{x \left(a - b x\right)}\, dx = \int 1\, dt$$
Detailed solution of the integral with x
Detailed solution of the integral with t
Take this integrals
$$- \frac{- \log{\left(x \right)} + \log{\left(- \frac{a}{b} + x \right)}}{a} = Const + t$$
Detailed solution of the equation
We get the simple equation with the unknown variable x.
(Const - it is a constant)
Solution is:
$$\operatorname{x_{1}} = x{\left(t \right)} = \frac{a e^{a \left(C_{1} + t\right)}}{b \left(e^{a \left(C_{1} + t\right)} - 1\right)}$$
The answer
[src]
a*(C1 + t)
a*e
x(t) = --------------------
/ a*(C1 + t)\
b*\-1 + e /
$$x{\left(t \right)} = \frac{a e^{a \left(C_{1} + t\right)}}{b \left(e^{a \left(C_{1} + t\right)} - 1\right)}$$
The classification
separable
1st power series
lie group
separable Integral