Internal
problem
ID
[8798]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
1.0
Problem
number
:
85
Date
solved
:
Friday, February 21, 2025 at 08:33:57 PM
CAS
classification
:
[_Bernoulli]
Solve
With initial conditions
This is non linear first order ODE. In canonical form it is written as
The \(x\) domain of \(f(x,y)\) when \(y=1\) is
And the point \(x_0 = 0\) is inside this domain. The \(y\) domain of \(f(x,y)\) when \(x=0\) is
And the point \(y_0 = 1\) is inside this domain. Now we will look at the continuity of
The \(x\) domain of \(\frac {\partial f}{\partial y}\) when \(y=1\) is
And the point \(x_0 = 0\) is inside this domain. The \(y\) domain of \(\frac {\partial f}{\partial y}\) when \(x=0\) is
And the point \(y_0 = 1\) is inside this domain. Therefore solution exists and is unique.
Time used: 0.195 (sec)
In canonical form, the ODE is
This is a Bernoulli ODE.
The standard Bernoulli ODE has the form
Comparing this to (1) shows that
The first step is to divide the above equation by \(y^n \) which gives
The next step is use the substitution \(v = y^{1-n}\) in equation (3) which generates a new ODE in \(v \left (x \right )\) which will be linear and can be easily solved using an integrating factor. Backsubstitution then gives the solution \(y(x)\) which is what we want.
This method is now applied to the ODE at hand. Comparing the ODE (1) With (2) Shows that
Dividing both sides of ODE (1) by \(y^n=y^{{3}/{2}}\) gives
Let
Taking derivative of equation (5) w.r.t \(x\) gives
Substituting equations (5) and (6) into equation (4) gives
The above now is a linear ODE in \(v \left (x \right )\) which is now solved.
In canonical form a linear first order is
Comparing the above to the given ode shows that
The integrating factor \(\mu \) is
The ode becomes
Integrating gives
Dividing throughout by the integrating factor \({\mathrm e}^{-x}\) gives the final solution
The substitution \(v = y^{1-n}\) is now used to convert the above solution back to \(y\) which results in
Solving for the constant of integration from initial conditions, the solution becomes
Solving for \(y\) gives
| |
Solution \(y = \frac {1}{\left (x +1\right )^{2}}\) | Slope field \(y^{\prime } = 2 y \left (x \sqrt {y}-1\right )\) |
Summary of solutions found
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
Solving time : 0.040
(sec)
Leaf size : 9
dsolve([diff(y(x),x) = 2*y(x)*(x*y(x)^(1/2)-1),op([y(0) = 1])],y(x),singsol=all)
Solving time : 0.682
(sec)
Leaf size : 20
DSolve[{D[y[x],x]==2*y[x]*(x*Sqrt[y[x]-1]),{y[0]==1}},y[x],x,IncludeSingularSolutions->True]
Solving time : 0.000
(sec)
Leaf size : 0
Python version: 3.13.1 (main, Dec 4 2024, 18:05:56) [GCC 14.2.1 20240910] Sympy version 1.13.3
from sympy import * x = symbols("x") y = Function("y") ode = Eq((-2*x*sqrt(y(x)) + 2)*y(x) + Derivative(y(x), x),0) ics = {y(0): 1} dsolve(ode,func=y(x),ics=ics)
NotImplementedError : Initial conditions produced too many solutions for constants