Internal problem ID [14184]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number: 26.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {-\frac {\left (1-\frac {1}{y}\right )^{2} y^{\prime }}{y^{2}}=-\frac {x -2}{x^{2}-4 x +3}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 977
dsolve((x-2)/(x^2-4*x+3)=(1-1/y(x))^2*1/y(x)^2*diff(y(x),x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (\left (9 \,\operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right ) \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )^{2}+36 \,\operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right ) \ln \left (\left (-1+x \right ) \left (-3+x \right )\right ) c_{1} +36 \,\operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right ) c_{1}^{2}-9 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )^{2}+6 \,\operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right ) \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )-36 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right ) c_{1} +12 \,\operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right ) c_{1} -36 c_{1}^{2}-18 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )-36 c_{1} -8\right )^{\frac {2}{3}}+6 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )-2 \,36^{\frac {1}{3}} \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {1}{3}}+12 c_{1} +4\right ) 36^{\frac {2}{3}}}{36 \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {1}{3}} \left (6 c_{1} +3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )\right )} \\ y \left (x \right ) &= \frac {\left (-i 36^{\frac {2}{3}} \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {2}{3}} \sqrt {3}+6 i \ln \left (\left (-1+x \right ) \left (-3+x \right )\right ) \sqrt {3}+12 i \sqrt {3}\, c_{1} +4 i \sqrt {3}-36^{\frac {2}{3}} \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {2}{3}}-6 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )-4 \,36^{\frac {1}{3}} \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {1}{3}}-12 c_{1} -4\right ) 36^{\frac {2}{3}}}{216 \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {1}{3}} \left (\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+2 c_{1} \right )} \\ y \left (x \right ) &= \frac {\left (i 36^{\frac {2}{3}} \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {2}{3}} \sqrt {3}-6 i \ln \left (\left (-1+x \right ) \left (-3+x \right )\right ) \sqrt {3}-12 i \sqrt {3}\, c_{1} -4 i \sqrt {3}-36^{\frac {2}{3}} \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {2}{3}}-6 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )-4 \,36^{\frac {1}{3}} \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {1}{3}}-12 c_{1} -4\right ) 36^{\frac {2}{3}}}{216 \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {1}{3}} \left (\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+2 c_{1} \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.679 (sec). Leaf size: 1134
DSolve[(x-2)/(x^2-4*x+3)==(1-1/y[x])^2*1/y[x]^2*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-6 \log (x-3)-6 \log (x-1)-\left (9 \log ^2(x-3)+9 \log ^2(x-1)+18 \log (x-3)+18 \log (x-3) \log (x-1)+18 \log (x-1)+3 \sqrt {\left (3 \log ^2(x-3)+3 \log ^2(x-1)+2 \log (x-3) (3 \log (x-1)+1+6 c_1)+2 (1+6 c_1) \log (x-1)+4 c_1 (1+3 c_1)\right ){}^2}+36 c_1 \log (x-3)+36 c_1 \log (x-1)+8+36 c_1{}^2+36 c_1\right ){}^{2/3}-2 \sqrt [3]{9 \log ^2(x-3)+9 \log ^2(x-1)+18 \log (x-3)+18 \log (x-3) \log (x-1)+18 \log (x-1)+3 \sqrt {\left (3 \log ^2(x-3)+3 \log ^2(x-1)+2 \log (x-3) (3 \log (x-1)+1+6 c_1)+2 (1+6 c_1) \log (x-1)+4 c_1 (1+3 c_1)\right ){}^2}+36 c_1 \log (x-3)+36 c_1 \log (x-1)+8+36 c_1{}^2+36 c_1}-4-12 c_1}{3 (\log (x-3)+\log (x-1)+2 c_1) \sqrt [3]{9 \log ^2(x-3)+9 \log ^2(x-1)+18 \log (x-3)+18 \log (x-3) \log (x-1)+18 \log (x-1)+3 \sqrt {\left (3 \log ^2(x-3)+3 \log ^2(x-1)+2 \log (x-3) (3 \log (x-1)+1+6 c_1)+2 (1+6 c_1) \log (x-1)+4 c_1 (1+3 c_1)\right ){}^2}+36 c_1 \log (x-3)+36 c_1 \log (x-1)+8+36 c_1{}^2+36 c_1}} \\ y(x)\to \frac {\frac {36 \left (1+i \sqrt {3}\right ) (3 \log (x-3)+3 \log (x-1)+2+6 c_1)}{\sqrt [3]{9 \log ^2(x-3)+9 \log ^2(x-1)+18 \log (x-3)+18 \log (x-3) \log (x-1)+18 \log (x-1)+3 \sqrt {\left (3 \log ^2(x-3)+3 \log ^2(x-1)+2 \log (x-3) (3 \log (x-1)+1+6 c_1)+2 (1+6 c_1) \log (x-1)+4 c_1 (1+3 c_1)\right ){}^2}+36 c_1 \log (x-3)+36 c_1 \log (x-1)+8+36 c_1{}^2+36 c_1}}+18 \left (1-i \sqrt {3}\right ) \sqrt [3]{9 \log ^2(x-3)+9 \log ^2(x-1)+18 \log (x-3)+18 \log (x-3) \log (x-1)+18 \log (x-1)+3 \sqrt {\left (3 \log ^2(x-3)+3 \log ^2(x-1)+2 \log (x-3) (3 \log (x-1)+1+6 c_1)+2 (1+6 c_1) \log (x-1)+4 c_1 (1+3 c_1)\right ){}^2}+36 c_1 \log (x-3)+36 c_1 \log (x-1)+8+36 c_1{}^2+36 c_1}-72}{108 (\log (x-3)+\log (x-1)+2 c_1)} \\ y(x)\to \frac {\frac {36 \left (1-i \sqrt {3}\right ) (3 \log (x-3)+3 \log (x-1)+2+6 c_1)}{\sqrt [3]{9 \log ^2(x-3)+9 \log ^2(x-1)+18 \log (x-3)+18 \log (x-3) \log (x-1)+18 \log (x-1)+3 \sqrt {\left (3 \log ^2(x-3)+3 \log ^2(x-1)+2 \log (x-3) (3 \log (x-1)+1+6 c_1)+2 (1+6 c_1) \log (x-1)+4 c_1 (1+3 c_1)\right ){}^2}+36 c_1 \log (x-3)+36 c_1 \log (x-1)+8+36 c_1{}^2+36 c_1}}+18 \left (1+i \sqrt {3}\right ) \sqrt [3]{9 \log ^2(x-3)+9 \log ^2(x-1)+18 \log (x-3)+18 \log (x-3) \log (x-1)+18 \log (x-1)+3 \sqrt {\left (3 \log ^2(x-3)+3 \log ^2(x-1)+2 \log (x-3) (3 \log (x-1)+1+6 c_1)+2 (1+6 c_1) \log (x-1)+4 c_1 (1+3 c_1)\right ){}^2}+36 c_1 \log (x-3)+36 c_1 \log (x-1)+8+36 c_1{}^2+36 c_1}-72}{108 (\log (x-3)+\log (x-1)+2 c_1)} \\ y(x)\to 0 \\ \end{align*}