Internal problem ID [5091]
Internal file name [OUTPUT/4584_Sunday_June_05_2022_03_01_20_PM_83982119/index.tex]
Book: Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY.
2001
Section: Program 24. First order differential equations. Further problems 24. page
1068
Problem number: 5.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {x^{2} \left (y+1\right )+y^{2} \left (-1+x \right ) y^{\prime }=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {x^{2} \left (y +1\right )}{y^{2} \left (-1+x \right )} \end {align*}
Where \(f(x)=-\frac {x^{2}}{-1+x}\) and \(g(y)=\frac {y +1}{y^{2}}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {y +1}{y^{2}}} \,dy &= -\frac {x^{2}}{-1+x} \,d x \\ \int { \frac {1}{\frac {y +1}{y^{2}}} \,dy} &= \int {-\frac {x^{2}}{-1+x} \,d x} \\ \frac {y^{2}}{2}-y +\ln \left (y +1\right )&=-\frac {x^{2}}{2}-x -\ln \left (-1+x \right )+c_{1} \\ \end{align*} Which results in \begin{align*} y &= {\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}}-x^{2}+4 \,{\mathrm e}^{\textit {\_Z}}-2 \ln \left (-1+x \right )+2 c_{1} -2 \textit {\_Z} -2 x -3\right )}-1 \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}}-x^{2}+4 \,{\mathrm e}^{\textit {\_Z}}-2 \ln \left (-1+x \right )+2 c_{1} -2 \textit {\_Z} -2 x -3\right )}-1 \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}}-x^{2}+4 \,{\mathrm e}^{\textit {\_Z}}-2 \ln \left (-1+x \right )+2 c_{1} -2 \textit {\_Z} -2 x -3\right )}-1 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (y+1\right )+y^{2} \left (-1+x \right ) y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {x^{2} \left (y+1\right )}{y^{2} \left (-1+x \right )} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } y^{2}}{y+1}=-\frac {x^{2}}{-1+x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } y^{2}}{y+1}d x =\int -\frac {x^{2}}{-1+x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -y+\frac {y^{2}}{2}+\ln \left (y+1\right )=-\frac {x^{2}}{2}-x -\ln \left (-1+x \right )+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 30
dsolve(x^2*(y(x)+1)+y(x)^2*(x-1)*diff(y(x),x)=0,y(x), singsol=all)
\[ \frac {x^{2}}{2}+x +\ln \left (x -1\right )+\frac {y \left (x \right )^{2}}{2}-y \left (x \right )+\ln \left (y \left (x \right )+1\right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.42 (sec). Leaf size: 56
DSolve[x^2*(y[x]+1)+y[x]^2*(x-1)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {1}{2} (\text {$\#$1}+1)^2-2 (\text {$\#$1}+1)+\log (\text {$\#$1}+1)\&\right ]\left [-\frac {x^2}{2}-x-\log (x-1)+\frac {3}{2}+c_1\right ] \\ y(x)\to -1 \\ \end{align*}