Internal problem ID [14117]
Internal file name [OUTPUT/13798_Saturday_March_02_2024_02_50_44_PM_58513738/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 1. Introduction to Differential Equations. Review exercises, page
23
Problem number: 18.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y^{\prime }=\frac {2 x^{2}-x +1}{\left (x -1\right ) \left (x^{2}+1\right )}} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {2 x^{2}-x +1}{\left (x -1\right ) \left (x^{2}+1\right )}\,\mathop {\mathrm {d}x}}\\ &= \ln \left (x -1\right )+\frac {\ln \left (x^{2}+1\right )}{2}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \ln \left (x -1\right )+\frac {\ln \left (x^{2}+1\right )}{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \ln \left (x -1\right )+\frac {\ln \left (x^{2}+1\right )}{2}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {2 x^{2}-x +1}{\left (x -1\right ) \left (x^{2}+1\right )} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {2 x^{2}-x +1}{\left (x -1\right ) \left (x^{2}+1\right )}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\ln \left (x -1\right )+\frac {\ln \left (x^{2}+1\right )}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\ln \left (x -1\right )+\frac {\ln \left (x^{2}+1\right )}{2}+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 18
dsolve(diff(y(x),x)=(2*x^2-x+1)/( (x-1)*(x^2+1)),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\ln \left (x^{2}+1\right )}{2}+\ln \left (-1+x \right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 24
DSolve[y'[x]==(2*x^2-x+1)/( (x-1)*(x^2+1)),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} \log \left (x^2+1\right )+\log (1-x)+c_1 \]