74.4.26 problem 26

Internal problem ID [15919]
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
Date solved : Tuesday, January 28, 2025 at 08:21:20 AM
CAS classification : [_separable]

\begin{align*} \frac {x -2}{x^{2}-4 x +3}&=\frac {\left (1-\frac {1}{y}\right )^{2} y^{\prime }}{y^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 969

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 &= \frac {\left (\left (9 \,\operatorname {csgn}\left (3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+6 c_{1} +2\right ) \ln \left (\left (x -1\right ) \left (x -3\right )\right )^{2}+36 \,\operatorname {csgn}\left (3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+6 c_{1} +2\right ) \ln \left (\left (x -1\right ) \left (x -3\right )\right ) c_{1} +36 \,\operatorname {csgn}\left (3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+6 c_{1} +2\right ) c_{1}^{2}-9 \ln \left (\left (x -1\right ) \left (x -3\right )\right )^{2}+6 \,\operatorname {csgn}\left (3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+6 c_{1} +2\right ) \ln \left (\left (x -1\right ) \left (x -3\right )\right )-36 \ln \left (\left (x -1\right ) \left (x -3\right )\right ) c_{1} +12 \,\operatorname {csgn}\left (3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+6 c_{1} +2\right ) c_{1} -36 c_{1}^{2}-18 \ln \left (\left (x -1\right ) \left (x -3\right )\right )-36 c_{1} -8\right )^{{2}/{3}}-2 \,36^{{1}/{3}} \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} +\frac {1}{3}\right ) \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}-\frac {2}{3}\right )\right )^{{1}/{3}}+6 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+12 c_{1} +4\right ) 36^{{2}/{3}}}{36 \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} +\frac {1}{3}\right ) \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}-\frac {2}{3}\right )\right )^{{1}/{3}} \left (6 c_{1} +3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )\right )} \\ y &= \frac {\left (-i 36^{{2}/{3}} \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} +\frac {1}{3}\right ) \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}-\frac {2}{3}\right )\right )^{{2}/{3}} \sqrt {3}+6 i \ln \left (\left (x -1\right ) \left (x -3\right )\right ) \sqrt {3}+12 i \sqrt {3}\, c_{1} +4 i \sqrt {3}-36^{{2}/{3}} \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} +\frac {1}{3}\right ) \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}-\frac {2}{3}\right )\right )^{{2}/{3}}-6 \ln \left (\left (x -1\right ) \left (x -3\right )\right )-12 c_{1} -4 \,36^{{1}/{3}} \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} +\frac {1}{3}\right ) \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}-\frac {2}{3}\right )\right )^{{1}/{3}}-4\right ) 36^{{2}/{3}}}{216 \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} +\frac {1}{3}\right ) \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}-\frac {2}{3}\right )\right )^{{1}/{3}} \left (\ln \left (\left (x -1\right ) \left (x -3\right )\right )+2 c_{1} \right )} \\ y &= \frac {\left (i 36^{{2}/{3}} \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} +\frac {1}{3}\right ) \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}-\frac {2}{3}\right )\right )^{{2}/{3}} \sqrt {3}-6 i \ln \left (\left (x -1\right ) \left (x -3\right )\right ) \sqrt {3}-12 i \sqrt {3}\, c_{1} -36^{{2}/{3}} \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} +\frac {1}{3}\right ) \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}-\frac {2}{3}\right )\right )^{{2}/{3}}-4 i \sqrt {3}-4 \,36^{{1}/{3}} \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} +\frac {1}{3}\right ) \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}-\frac {2}{3}\right )\right )^{{1}/{3}}-6 \ln \left (\left (x -1\right ) \left (x -3\right )\right )-12 c_{1} -4\right ) 36^{{2}/{3}}}{216 \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} +\frac {1}{3}\right ) \left (\left (\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}-\frac {2}{3}\right )\right )^{{1}/{3}} \left (\ln \left (\left (x -1\right ) \left (x -3\right )\right )+2 c_{1} \right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 10.925 (sec). Leaf size: 1539

DSolve[(x-2)/(x^2-4*x+3)==(1-1/y[x])^2*1/y[x]^2*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {-6 \sqrt [3]{2} \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]-2^{2/3} \left (9 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]{}^2+9 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+18 c_1 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+3 \sqrt {\left (3 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]{}^2+(1+6 c_1) \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+c_1 (1+3 c_1)\right ){}^2}+2+9 c_1{}^2+9 c_1\right ){}^{2/3}-2 \sqrt [3]{9 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]{}^2+9 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+18 c_1 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+3 \sqrt {\left (3 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]{}^2+(1+6 c_1) \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+c_1 (1+3 c_1)\right ){}^2}+2+9 c_1{}^2+9 c_1}-2 \sqrt [3]{2}-6 \sqrt [3]{2} c_1}{6 \left (\int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+c_1\right ) \sqrt [3]{9 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]{}^2+9 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+18 c_1 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+3 \sqrt {\left (3 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]{}^2+(1+6 c_1) \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+c_1 (1+3 c_1)\right ){}^2}+2+9 c_1{}^2+9 c_1}} \\ y(x)\to \frac {\frac {18 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) \left (3 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+1+3 c_1\right )}{\sqrt [3]{9 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]{}^2+9 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+18 c_1 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+3 \sqrt {\left (3 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]{}^2+(1+6 c_1) \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+c_1 (1+3 c_1)\right ){}^2}+2+9 c_1{}^2+9 c_1}}+9\ 2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{9 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]{}^2+9 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+18 c_1 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+3 \sqrt {\left (3 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]{}^2+(1+6 c_1) \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+c_1 (1+3 c_1)\right ){}^2}+2+9 c_1{}^2+9 c_1}-36}{108 \left (\int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+c_1\right )} \\ y(x)\to \frac {\frac {18 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) \left (3 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+1+3 c_1\right )}{\sqrt [3]{9 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]{}^2+9 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+18 c_1 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+3 \sqrt {\left (3 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]{}^2+(1+6 c_1) \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+c_1 (1+3 c_1)\right ){}^2}+2+9 c_1{}^2+9 c_1}}+9\ 2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{9 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]{}^2+9 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+18 c_1 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+3 \sqrt {\left (3 \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]{}^2+(1+6 c_1) \int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+c_1 (1+3 c_1)\right ){}^2}+2+9 c_1{}^2+9 c_1}-36}{108 \left (\int _1^x\frac {K[1]-2}{(K[1]-3) (K[1]-1)}dK[1]+c_1\right )} \\ y(x)\to 0 \\ \end{align*}