3.13 problem 14

Internal problem ID [940]

Book: Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 2, First order equations. separable equations. Section 2.2 Page 52
Problem number: 14.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [_separable]

\[ \boxed {y^{\prime }+\frac {\left (1+y\right ) \left (y-1\right ) \left (y-2\right )}{x +1}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}

Solution by Maple

Time used: 6.86 (sec). Leaf size: 633

dsolve([diff(y(x),x)+((y(x)+1)*(y(x)-1)*(y(x)-2))/(x+1)=0,y(1) = 0],y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {8 \operatorname {RootOf}\left (512 \textit {\_Z}^{6}+\left (96 \,2^{\frac {1}{3}} x^{2}+192 x 2^{\frac {1}{3}}+96 \,2^{\frac {1}{3}}\right ) \textit {\_Z}^{4}-x^{6}-6 x^{5}-15 x^{4}-20 x^{3}-15 x^{2}-6 x -257\right )^{2}-2^{\frac {1}{3}} \left (x +1\right )^{2}}{8 \operatorname {RootOf}\left (512 \textit {\_Z}^{6}+\left (96 \,2^{\frac {1}{3}} x^{2}+192 x 2^{\frac {1}{3}}+96 \,2^{\frac {1}{3}}\right ) \textit {\_Z}^{4}-x^{6}-6 x^{5}-15 x^{4}-20 x^{3}-15 x^{2}-6 x -257\right )^{2}} \\ y \left (x \right ) &= \frac {16 \operatorname {RootOf}\left (512 \textit {\_Z}^{6}+\left (48 i x^{2} \sqrt {3}\, 2^{\frac {1}{3}}+96 i x \sqrt {3}\, 2^{\frac {1}{3}}+48 i \sqrt {3}\, 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}} x^{2}-96 x 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}}\right ) \textit {\_Z}^{4}-x^{6}-6 x^{5}-15 x^{4}-20 x^{3}-15 x^{2}-6 x -257\right )^{2}-\left (x +1\right )^{2} \left (i \sqrt {3}-1\right ) 2^{\frac {1}{3}}}{16 \operatorname {RootOf}\left (512 \textit {\_Z}^{6}+\left (48 i x^{2} \sqrt {3}\, 2^{\frac {1}{3}}+96 i x \sqrt {3}\, 2^{\frac {1}{3}}+48 i \sqrt {3}\, 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}} x^{2}-96 x 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}}\right ) \textit {\_Z}^{4}-x^{6}-6 x^{5}-15 x^{4}-20 x^{3}-15 x^{2}-6 x -257\right )^{2}} \\ y \left (x \right ) &= \frac {16 \operatorname {RootOf}\left (512 \textit {\_Z}^{6}+\left (-48 i x^{2} \sqrt {3}\, 2^{\frac {1}{3}}-96 i x \sqrt {3}\, 2^{\frac {1}{3}}-48 i \sqrt {3}\, 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}} x^{2}-96 x 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}}\right ) \textit {\_Z}^{4}-x^{6}-6 x^{5}-15 x^{4}-20 x^{3}-15 x^{2}-6 x -257\right )^{2}+\left (1+i \sqrt {3}\right ) \left (x +1\right )^{2} 2^{\frac {1}{3}}}{16 \operatorname {RootOf}\left (512 \textit {\_Z}^{6}+\left (-48 i x^{2} \sqrt {3}\, 2^{\frac {1}{3}}-96 i x \sqrt {3}\, 2^{\frac {1}{3}}-48 i \sqrt {3}\, 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}} x^{2}-96 x 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}}\right ) \textit {\_Z}^{4}-x^{6}-6 x^{5}-15 x^{4}-20 x^{3}-15 x^{2}-6 x -257\right )^{2}} \\ \end{align*}

Solution by Mathematica

Time used: 60.912 (sec). Leaf size: 1618

DSolve[{y'[x]+((y[x]+1)*(y[x]-1)*(y[x]-2))/(x+1)==0,y[1]==0},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\left (-1-i \sqrt {3}\right ) x^{12}-12 i \left (-i+\sqrt {3}\right ) x^{11}-66 i \left (-i+\sqrt {3}\right ) x^{10}-220 i \left (-i+\sqrt {3}\right ) x^9-495 i \left (-i+\sqrt {3}\right ) x^8-792 i \left (-i+\sqrt {3}\right ) x^7+2 \left (\sqrt [3]{-x^{18}-18 x^{17}-153 x^{16}-816 x^{15}-3060 x^{14}-8568 x^{13}-19076 x^{12}-37968 x^{11}-77550 x^{10}-161260 x^9-297198 x^8-437328 x^7-557188 x^6-807288 x^5-1239540 x^4-1424176 x^3-1016985 x^2-399378 x+16 \sqrt {(x+1)^{12} \left (x^6+6 x^5+15 x^4+20 x^3+15 x^2+6 x+257\right )^3}-66049}-590 i \sqrt {3}-590\right ) x^6+12 \left (\sqrt [3]{-x^{18}-18 x^{17}-153 x^{16}-816 x^{15}-3060 x^{14}-8568 x^{13}-19076 x^{12}-37968 x^{11}-77550 x^{10}-161260 x^9-297198 x^8-437328 x^7-557188 x^6-807288 x^5-1239540 x^4-1424176 x^3-1016985 x^2-399378 x+16 \sqrt {(x+1)^{12} \left (x^6+6 x^5+15 x^4+20 x^3+15 x^2+6 x+257\right )^3}-66049}-194 i \sqrt {3}-194\right ) x^5+15 \left (2 \sqrt [3]{-x^{18}-18 x^{17}-153 x^{16}-816 x^{15}-3060 x^{14}-8568 x^{13}-19076 x^{12}-37968 x^{11}-77550 x^{10}-161260 x^9-297198 x^8-437328 x^7-557188 x^6-807288 x^5-1239540 x^4-1424176 x^3-1016985 x^2-399378 x+16 \sqrt {(x+1)^{12} \left (x^6+6 x^5+15 x^4+20 x^3+15 x^2+6 x+257\right )^3}-66049}-289 i \sqrt {3}-289\right ) x^4+20 \left (2 \sqrt [3]{-x^{18}-18 x^{17}-153 x^{16}-816 x^{15}-3060 x^{14}-8568 x^{13}-19076 x^{12}-37968 x^{11}-77550 x^{10}-161260 x^9-297198 x^8-437328 x^7-557188 x^6-807288 x^5-1239540 x^4-1424176 x^3-1016985 x^2-399378 x+16 \sqrt {(x+1)^{12} \left (x^6+6 x^5+15 x^4+20 x^3+15 x^2+6 x+257\right )^3}-66049}-267 i \sqrt {3}-267\right ) x^3+6 \left (5 \sqrt [3]{-x^{18}-18 x^{17}-153 x^{16}-816 x^{15}-3060 x^{14}-8568 x^{13}-19076 x^{12}-37968 x^{11}-77550 x^{10}-161260 x^9-297198 x^8-437328 x^7-557188 x^6-807288 x^5-1239540 x^4-1424176 x^3-1016985 x^2-399378 x+16 \sqrt {(x+1)^{12} \left (x^6+6 x^5+15 x^4+20 x^3+15 x^2+6 x+257\right )^3}-66049}-651 i \sqrt {3}-651\right ) x^2+12 \left (\sqrt [3]{-x^{18}-18 x^{17}-153 x^{16}-816 x^{15}-3060 x^{14}-8568 x^{13}-19076 x^{12}-37968 x^{11}-77550 x^{10}-161260 x^9-297198 x^8-437328 x^7-557188 x^6-807288 x^5-1239540 x^4-1424176 x^3-1016985 x^2-399378 x+16 \sqrt {(x+1)^{12} \left (x^6+6 x^5+15 x^4+20 x^3+15 x^2+6 x+257\right )^3}-66049}-129 i \sqrt {3}-129\right ) x+i \sqrt {3} \left (-x^{18}-18 x^{17}-153 x^{16}-816 x^{15}-3060 x^{14}-8568 x^{13}-19076 x^{12}-37968 x^{11}-77550 x^{10}-161260 x^9-297198 x^8-437328 x^7-557188 x^6-807288 x^5-1239540 x^4-1424176 x^3-1016985 x^2-399378 x+16 \sqrt {(x+1)^{12} \left (x^6+6 x^5+15 x^4+20 x^3+15 x^2+6 x+257\right )^3}-66049\right )^{2/3}-\left (-x^{18}-18 x^{17}-153 x^{16}-816 x^{15}-3060 x^{14}-8568 x^{13}-19076 x^{12}-37968 x^{11}-77550 x^{10}-161260 x^9-297198 x^8-437328 x^7-557188 x^6-807288 x^5-1239540 x^4-1424176 x^3-1016985 x^2-399378 x+16 \sqrt {(x+1)^{12} \left (x^6+6 x^5+15 x^4+20 x^3+15 x^2+6 x+257\right )^3}-66049\right )^{2/3}+514 \sqrt [3]{-x^{18}-18 x^{17}-153 x^{16}-816 x^{15}-3060 x^{14}-8568 x^{13}-19076 x^{12}-37968 x^{11}-77550 x^{10}-161260 x^9-297198 x^8-437328 x^7-557188 x^6-807288 x^5-1239540 x^4-1424176 x^3-1016985 x^2-399378 x+16 \sqrt {(x+1)^{12} \left (x^6+6 x^5+15 x^4+20 x^3+15 x^2+6 x+257\right )^3}-66049}-257 i \sqrt {3}-257}{2 \left (x^6+6 x^5+15 x^4+20 x^3+15 x^2+6 x+257\right ) \sqrt [3]{-x^{18}-18 x^{17}-153 x^{16}-816 x^{15}-3060 x^{14}-8568 x^{13}-19076 x^{12}-37968 x^{11}-77550 x^{10}-161260 x^9-297198 x^8-437328 x^7-557188 x^6-807288 x^5-1239540 x^4-1424176 x^3-1016985 x^2-399378 x+16 \sqrt {(x+1)^{12} \left (x^6+6 x^5+15 x^4+20 x^3+15 x^2+6 x+257\right )^3}-66049}} \]