Internal problem ID [13864]
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: 0.0 (sec). Leaf size: 1790
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 (-9 \ln \left (\left (x -1\right ) \left (x -3\right )\right )^{2}+3 \sqrt {9 \ln \left (\left (x -1\right ) \left (x -3\right )\right )^{2}+36 \ln \left (\left (x -1\right ) \left (x -3\right )\right ) c_{1} +36 c_{1}^{2}+12 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+24 c_{1} +4}\, \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} +6 \sqrt {9 \ln \left (\left (x -1\right ) \left (x -3\right )\right )^{2}+36 \ln \left (\left (x -1\right ) \left (x -3\right )\right ) c_{1} +36 c_{1}^{2}+12 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+24 c_{1} +4}\, c_{1} -36 c_{1}^{2}-18 \ln \left (\left (x -1\right ) \left (x -3\right )\right )-36 c_{1} -8\right )^{\frac {1}{3}}}{6 c_{1} +3 \ln \left (\left (x -1\right ) \left (x -3\right )\right )}+\frac {2 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+4 c_{1} +\frac {4}{3}}{\left (\ln \left (\left (x -1\right ) \left (x -3\right )\right )+2 c_{1} \right ) \left (-9 \ln \left (\left (x -1\right ) \left (x -3\right )\right )^{2}+3 \sqrt {9 \ln \left (\left (x -1\right ) \left (x -3\right )\right )^{2}+36 \ln \left (\left (x -1\right ) \left (x -3\right )\right ) c_{1} +36 c_{1}^{2}+12 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+24 c_{1} +4}\, \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} +6 \sqrt {9 \ln \left (\left (x -1\right ) \left (x -3\right )\right )^{2}+36 \ln \left (\left (x -1\right ) \left (x -3\right )\right ) c_{1} +36 c_{1}^{2}+12 \ln \left (\left (x -1\right ) \left (x -3\right )\right )+24 c_{1} +4}\, c_{1} -36 c_{1}^{2}-18 \ln \left (\left (x -1\right ) \left (x -3\right )\right )-36 c_{1} -8\right )^{\frac {1}{3}}}-\frac {2}{3 \left (\ln \left (\left (x -1\right ) \left (x -3\right )\right )+2 c_{1} \right )} \text {Expression too large to display} \text {Expression too large to display} \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*}