Mister Exam
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
-
How to use it?
- Differential equation:
- Equation y'-2y=3e^2x
- Equation dy/dx=xy^(1/2)
- Equation y'+y*sinx=e^(cosx)
- Equation x^2dx+y(x-1)dy=0
- Derivative of:
- dy/dx
-
Identical expressions
- dy/dx=xy^(one / two)
- dy divide by dx equally xy to the power of (1 divide by 2)
- dy divide by dx equally xy to the power of (one divide by two)
- dy/dx=xy(1/2)
- dy/dx=xy1/2
- dy/dx=xy^1/2
- dy divide by dx=xy^(1 divide by 2)
-
Expressions with functions
- dy
- dy/dx=4y-8
- dy/dx=sin(y)
- dy/dx=y^(2/3)
- dy/dx=ycosx/1+2y^2
- dy/dx=(x^2+3y^2)/(2xy)
- dx
- dx/dt=3xt^2
- dx/dt+3x/20+t=4
- dx/dt=-k(a-x)
- dx/dt=x
- dx/(xz-y)=dy/(yz-x)=dz/(xy-z)
- xy
- xy''-(2x+1)y'+(x+1)y=0
- xydx-(x^2+3y^2)dy=0
- xy'=y-xe^(y/x)
- xydx+2(x^2+2y^2)dy=0
- xydx-(x^2+2y^2)dy=0
Differential equation dy/dx=xy^(1/2)
⌨
The solution
You have entered
[src]
d ______ --(y(x)) = x*\/ y(x) dx
$$\frac{d}{d x} y{\left(x \right)} = x \sqrt{y{\left(x \right)}}$$
y' = x*sqrt(y)
Detail solution
Given the equation:
$$\frac{d}{d x} y{\left(x \right)} = x \sqrt{y{\left(x \right)}}$$
This differential equation has the form:
where
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = x$$
$$\operatorname{g_{2}}{\left(y \right)} = \sqrt{y{\left(x \right)}}$$
We give the equation to the form:
Divide both parts of the equation by g2(y)
$$\sqrt{y{\left(x \right)}}$$
we get
$$\frac{\frac{d}{d x} y{\left(x \right)}}{\sqrt{y{\left(x \right)}}} = x$$
We separated the variables x and y.
Now, multiply the both equation sides by dx,
then the equation will be the
$$\frac{dx \frac{d}{d x} y{\left(x \right)}}{\sqrt{y{\left(x \right)}}} = dx x$$
or
$$\frac{dy}{\sqrt{y{\left(x \right)}}} = dx x$$
Take the integrals from the both equation sides:
- the integral of the left side by y,
- the integral of the right side by x.
$$\int \frac{1}{\sqrt{y}}\, dy = \int x\, dx$$
Detailed solution of the integral with y
Detailed solution of the integral with x
Take this integrals
$$2 \sqrt{y} = Const + \frac{x^{2}}{2}$$
Detailed solution of the equation
We get the simple equation with the unknown variable y.
(Const - it is a constant)
Solution is:
$$\operatorname{y_{1}} = y{\left(x \right)} = \frac{C_{1}^{2}}{4} + \frac{C_{1} x^{2}}{4} + \frac{x^{4}}{16}$$
$$\frac{d}{d x} y{\left(x \right)} = x \sqrt{y{\left(x \right)}}$$
This differential equation has the form:
f1(x)*g1(y)*y' = f2(x)*g2(y),
where
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = x$$
$$\operatorname{g_{2}}{\left(y \right)} = \sqrt{y{\left(x \right)}}$$
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)
$$\sqrt{y{\left(x \right)}}$$
we get
$$\frac{\frac{d}{d x} y{\left(x \right)}}{\sqrt{y{\left(x \right)}}} = x$$
We separated the variables x and y.
Now, multiply the both equation sides by dx,
then the equation will be the
$$\frac{dx \frac{d}{d x} y{\left(x \right)}}{\sqrt{y{\left(x \right)}}} = dx x$$
or
$$\frac{dy}{\sqrt{y{\left(x \right)}}} = dx x$$
Take the integrals from the both equation sides:
- the integral of the left side by y,
- the integral of the right side by x.
$$\int \frac{1}{\sqrt{y}}\, dy = \int x\, dx$$
Detailed solution of the integral with y
Detailed solution of the integral with x
Take this integrals
$$2 \sqrt{y} = Const + \frac{x^{2}}{2}$$
Detailed solution of the equation
We get the simple equation with the unknown variable y.
(Const - it is a constant)
Solution is:
$$\operatorname{y_{1}} = y{\left(x \right)} = \frac{C_{1}^{2}}{4} + \frac{C_{1} x^{2}}{4} + \frac{x^{4}}{16}$$
The answer
[src]
2 4 2
C1 x C1*x
y(x) = --- + -- + -----
4 16 4
$$y{\left(x \right)} = \frac{C_{1}^{2}}{4} + \frac{C_{1} x^{2}}{4} + \frac{x^{4}}{16}$$
The classification
separable
1st exact
Bernoulli
1st power series
lie group
separable Integral
1st exact Integral
Bernoulli Integral
The graph