# Differential equation ydy=-xdx

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

from to

### The solution

You have entered [src]
d
--(y(x))*y(x) = -x
dx                
$$y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = - x$$
y*y' = -x
Detail solution
Given the equation:
$$y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = - x$$
This differential equation has the form:
$$f_1(x)*g_1(y)*y' = f_2(x)*g_2(y)$$
where
$$f_{1}{\left(x \right)} = 1$$
$$g_{1}{\left(y \right)} = 1$$
$$f_{2}{\left(x \right)} = - x$$
$$g_{2}{\left(y \right)} = \frac{1}{y{\left(x \right)}}$$
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 $g_{2}{\left(y{\left(x \right)} \right)}$
$$\frac{1}{y{\left(x \right)}}$$
we get
$$y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = - x$$
We separated the variables x and y.

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

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

Solution is:
$$y_{1} = y{\left(x \right)} = - \sqrt{- x^{2} + C_{1}}$$
$$y_{2} = y{\left(x \right)} = \sqrt{- x^{2} + C_{1}}$$
           _________
/       2
y(x) = -\/  C1 - x  
$$y{\left(x \right)} = - \sqrt{- x^{2} + C_{1}}$$
          _________
/       2
y(x) = \/  C1 - x  
$$y{\left(x \right)} = \sqrt{- x^{2} + C_{1}}$$
$${\it \%a}$$
a
Graph of the Cauchy problem
The classification
1st exact
1st exact Integral
1st homogeneous coeff best
1st homogeneous coeff subs dep div indep
1st homogeneous coeff subs dep div indep Integral
1st homogeneous coeff subs indep div dep
1st homogeneous coeff subs indep div dep Integral
1st power series
Bernoulli
Bernoulli Integral
factorable
lie group
separable
separable Integral
(x, y):
(-10.0, 0.75)
(-7.777777777777778, 6.329982240581089)
(-5.555555555555555, 8.348551178571897)
(-3.333333333333333, 9.457874778337375)
(-1.1111111111111107, 9.966340355961792)
(1.1111111111111107, 9.96634039631456)
(3.333333333333334, 9.457874568925822)
(5.555555555555557, 8.348550566706328)
(7.777777777777779, 6.329980785921501)
(10.0, 0.7499815578513588)
(10.0, 0.7499815578513588)
The graph To see a detailed solution - share to all your student friends
To see a detailed solution,
share to all your student friends: