Internal problem ID [5728]
Internal file name [OUTPUT/4976_Sunday_June_05_2022_03_15_42_PM_72193388/index.tex]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.1 Separable equations problems.
page 7
Problem number: 15.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {\frac {y}{x -1}+\frac {x y^{\prime }}{y+1}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {y \left (y +1\right )}{\left (x -1\right ) x} \end {align*}
Where \(f(x)=-\frac {1}{\left (x -1\right ) x}\) and \(g(y)=y \left (y +1\right )\). Integrating both sides gives \begin{align*} \frac {1}{y \left (y +1\right )} \,dy &= -\frac {1}{\left (x -1\right ) x} \,d x \\ \int { \frac {1}{y \left (y +1\right )} \,dy} &= \int {-\frac {1}{\left (x -1\right ) x} \,d x} \\ \ln \left (y \right )-\ln \left (y +1\right )&=\ln \left (x \right )-\ln \left (x -1\right )+c_{1} \\ \end{align*} Raising both side to exponential gives \begin {align*} {\mathrm e}^{\ln \left (y \right )-\ln \left (y +1\right )} &= {\mathrm e}^{\ln \left (x \right )-\ln \left (x -1\right )+c_{1}} \end {align*}
Which simplifies to \begin {align*} \frac {y}{y +1} &= c_{2} {\mathrm e}^{\ln \left (x \right )-\ln \left (x -1\right )} \end {align*}
Which simplifies to \[ y = -\frac {c_{2} x}{\left (x -1\right ) \left (-1+\frac {c_{2} x}{x -1}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {c_{2} x}{\left (x -1\right ) \left (-1+\frac {c_{2} x}{x -1}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {c_{2} x}{\left (x -1\right ) \left (-1+\frac {c_{2} x}{x -1}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {y}{x -1}+\frac {x y^{\prime }}{y+1}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {y \left (y+1\right )}{\left (x -1\right ) x} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y \left (y+1\right )}=-\frac {1}{\left (x -1\right ) x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y \left (y+1\right )}d x =\int -\frac {1}{\left (x -1\right ) x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\ln \left (y+1\right )+\ln \left (y\right )=\ln \left (x \right )-\ln \left (x -1\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {x \,{\mathrm e}^{c_{1}}}{1+x \,{\mathrm e}^{c_{1}}-x} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 15
dsolve(y(x)/(x-1)+x/(y(x)+1)*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x}{-1+c_{1} \left (x -1\right )} \]
✓ Solution by Mathematica
Time used: 0.417 (sec). Leaf size: 33
DSolve[y[x]/(x-1)+x/(y[x]+1)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {e^{c_1} x}{x+e^{c_1} x-1} \\ y(x)\to -1 \\ y(x)\to 0 \\ \end{align*}