74.7.56 problem 61

Internal problem ID [16133]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.5, page 64
Problem number : 61
Date solved : Tuesday, January 28, 2025 at 08:50:36 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

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

✓ Solution by Maple

Time used: 0.096 (sec). Leaf size: 666

dsolve(y(t)=t*(2-diff(y(t),t))+(2*diff(y(t),t)^2+1),y(t), singsol=all)
 

\begin{align*} y &= -\frac {t \left (\frac {t -4}{\left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{{1}/{3}}}+\left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{{1}/{3}}-2\right ) \left (\frac {t -4}{\left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{{1}/{3}}}+\left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{{1}/{3}}+2\right )}{4}+\frac {\left (\frac {\left (t -4\right )^{2}}{\left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{{2}/{3}}}+2 t -4+\left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{{2}/{3}}\right )^{2}}{8}+1 \\ y &= -\frac {\left (1+i \sqrt {3}\right ) \left (t^{3}-6 c_{1} \sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}+18 c_{1}^{2}-12 t^{2}+48 t -64\right ) \left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{{2}/{3}}+12 \left (c_{1} -\frac {\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}}{3}\right ) \left (t^{2}-4 t +28\right ) \left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{{1}/{3}}+\left (t^{3}-12 t^{2}-36 c_{1}^{2}+12 c_{1} \sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}+48 t -64\right ) \left (t -4\right ) \left (i \sqrt {3}-1\right )}{16 \left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{{4}/{3}}} \\ y &= \frac {\left (t^{3}-6 c_{1} \sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}+18 c_{1}^{2}-12 t^{2}+48 t -64\right ) \left (i \sqrt {3}-1\right ) \left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{{2}/{3}}-12 \left (c_{1} -\frac {\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}}{3}\right ) \left (t^{2}-4 t +28\right ) \left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{{1}/{3}}+\left (1+i \sqrt {3}\right ) \left (t^{3}-12 t^{2}-36 c_{1}^{2}+12 c_{1} \sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}+48 t -64\right ) \left (t -4\right )}{16 \left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{{4}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 61.106 (sec). Leaf size: 2369

DSolve[y[t]==t*(2-D[y[t],t])+(2*D[y[t],t]^2+1),y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to \frac {2 \sqrt [3]{2} t^4-32 \sqrt [3]{2} t^3+192 \sqrt [3]{2} t^2+4 t^2 \sqrt [3]{-2 t^6+48 t^5-480 t^4+2560 t^3-10 e^{3 c_1} t^3-7680 t^2+120 e^{3 c_1} t^2+12288 t-480 e^{3 c_1} t+\sqrt {e^{3 c_1} \left (4 (t-4)^3+e^{3 c_1}\right ){}^3}-8192+640 e^{3 c_1}+e^{6 c_1}}-16 t \sqrt [3]{-2 t^6+48 t^5-480 t^4+2560 t^3-10 e^{3 c_1} t^3-7680 t^2+120 e^{3 c_1} t^2+12288 t-480 e^{3 c_1} t+\sqrt {e^{3 c_1} \left (4 (t-4)^3+e^{3 c_1}\right ){}^3}-8192+640 e^{3 c_1}+e^{6 c_1}}+2^{2/3} \left (-2 t^6+48 t^5-480 t^4+2560 t^3-10 e^{3 c_1} t^3-7680 t^2+120 e^{3 c_1} t^2+12288 t-480 e^{3 c_1} t+\sqrt {e^{3 c_1} \left (4 (t-4)^3+e^{3 c_1}\right ){}^3}-8192+640 e^{3 c_1}+e^{6 c_1}\right ){}^{2/3}+112 \sqrt [3]{-2 t^6+48 t^5-480 t^4+2560 t^3-10 e^{3 c_1} t^3-7680 t^2+120 e^{3 c_1} t^2+12288 t-480 e^{3 c_1} t+\sqrt {e^{3 c_1} \left (4 (t-4)^3+e^{3 c_1}\right ){}^3}-8192+640 e^{3 c_1}+e^{6 c_1}}-512 \sqrt [3]{2} t-4 \sqrt [3]{2} e^{3 c_1} t+512 \sqrt [3]{2}+16 \sqrt [3]{2} e^{3 c_1}}{16 \sqrt [3]{-2 t^6+48 t^5-480 t^4+2560 t^3-10 e^{3 c_1} t^3-7680 t^2+120 e^{3 c_1} t^2+12288 t-480 e^{3 c_1} t+\sqrt {e^{3 c_1} \left (4 (t-4)^3+e^{3 c_1}\right ){}^3}-8192+640 e^{3 c_1}+e^{6 c_1}}} \\ y(t)\to \frac {t^2}{4}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-2 t^6+48 t^5-480 t^4+2560 t^3-10 e^{3 c_1} t^3-7680 t^2+120 e^{3 c_1} t^2+12288 t-480 e^{3 c_1} t+\sqrt {e^{3 c_1} \left (4 (t-4)^3+e^{3 c_1}\right ){}^3}-8192+640 e^{3 c_1}+e^{6 c_1}}}{16 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) (t-4) \left (-(t-4)^3+2 e^{3 c_1}\right )}{8\ 2^{2/3} \sqrt [3]{-2 t^6+48 t^5-480 t^4+2560 t^3-10 e^{3 c_1} t^3-7680 t^2+120 e^{3 c_1} t^2+12288 t-480 e^{3 c_1} t+\sqrt {e^{3 c_1} \left (4 (t-4)^3+e^{3 c_1}\right ){}^3}-8192+640 e^{3 c_1}+e^{6 c_1}}}-t+7 \\ y(t)\to \frac {t^2}{4}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{-2 t^6+48 t^5-480 t^4+2560 t^3-10 e^{3 c_1} t^3-7680 t^2+120 e^{3 c_1} t^2+12288 t-480 e^{3 c_1} t+\sqrt {e^{3 c_1} \left (4 (t-4)^3+e^{3 c_1}\right ){}^3}-8192+640 e^{3 c_1}+e^{6 c_1}}}{16 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) (t-4) \left ((t-4)^3-2 e^{3 c_1}\right )}{8\ 2^{2/3} \sqrt [3]{-2 t^6+48 t^5-480 t^4+2560 t^3-10 e^{3 c_1} t^3-7680 t^2+120 e^{3 c_1} t^2+12288 t-480 e^{3 c_1} t+\sqrt {e^{3 c_1} \left (4 (t-4)^3+e^{3 c_1}\right ){}^3}-8192+640 e^{3 c_1}+e^{6 c_1}}}-t+7 \\ y(t)\to \frac {t^4-16 t^3+96 t^2+2 t^2 \sqrt [3]{-t^6+24 t^5-240 t^4+1280 t^3+20 e^{3 c_1} t^3-3840 t^2-240 e^{3 c_1} t^2+6144 t+960 e^{3 c_1} t+8 \sqrt {e^{3 c_1} \left (-(t-4)^3+e^{3 c_1}\right ){}^3}-4096-1280 e^{3 c_1}+8 e^{6 c_1}}-8 t \sqrt [3]{-t^6+24 t^5-240 t^4+1280 t^3+20 e^{3 c_1} t^3-3840 t^2-240 e^{3 c_1} t^2+6144 t+960 e^{3 c_1} t+8 \sqrt {e^{3 c_1} \left (-(t-4)^3+e^{3 c_1}\right ){}^3}-4096-1280 e^{3 c_1}+8 e^{6 c_1}}+\left (-t^6+24 t^5-240 t^4+1280 t^3+20 e^{3 c_1} t^3-3840 t^2-240 e^{3 c_1} t^2+6144 t+960 e^{3 c_1} t+8 \sqrt {e^{3 c_1} \left (-(t-4)^3+e^{3 c_1}\right ){}^3}-4096-1280 e^{3 c_1}+8 e^{6 c_1}\right ){}^{2/3}+56 \sqrt [3]{-t^6+24 t^5-240 t^4+1280 t^3+20 e^{3 c_1} t^3-3840 t^2-240 e^{3 c_1} t^2+6144 t+960 e^{3 c_1} t+8 \sqrt {e^{3 c_1} \left (-(t-4)^3+e^{3 c_1}\right ){}^3}-4096-1280 e^{3 c_1}+8 e^{6 c_1}}-256 t+8 e^{3 c_1} t+256-32 e^{3 c_1}}{8 \sqrt [3]{-t^6+24 t^5-240 t^4+1280 t^3+20 e^{3 c_1} t^3-3840 t^2-240 e^{3 c_1} t^2+6144 t+960 e^{3 c_1} t+8 \sqrt {e^{3 c_1} \left (-(t-4)^3+e^{3 c_1}\right ){}^3}-4096-1280 e^{3 c_1}+8 e^{6 c_1}}} \\ y(t)\to \frac {t^2}{4}+\frac {1}{16} i \left (\sqrt {3}+i\right ) \sqrt [3]{-t^6+24 t^5-240 t^4+1280 t^3+20 e^{3 c_1} t^3-3840 t^2-240 e^{3 c_1} t^2+6144 t+960 e^{3 c_1} t+8 \sqrt {e^{3 c_1} \left (-(t-4)^3+e^{3 c_1}\right ){}^3}-4096-1280 e^{3 c_1}+8 e^{6 c_1}}-\frac {i \left (\sqrt {3}-i\right ) (t-4) \left ((t-4)^3+8 e^{3 c_1}\right )}{16 \sqrt [3]{-t^6+24 t^5-240 t^4+1280 t^3+20 e^{3 c_1} t^3-3840 t^2-240 e^{3 c_1} t^2+6144 t+960 e^{3 c_1} t+8 \sqrt {e^{3 c_1} \left (-(t-4)^3+e^{3 c_1}\right ){}^3}-4096-1280 e^{3 c_1}+8 e^{6 c_1}}}-t+7 \\ y(t)\to \frac {t^2}{4}-\frac {1}{16} i \left (\sqrt {3}-i\right ) \sqrt [3]{-t^6+24 t^5-240 t^4+1280 t^3+20 e^{3 c_1} t^3-3840 t^2-240 e^{3 c_1} t^2+6144 t+960 e^{3 c_1} t+8 \sqrt {e^{3 c_1} \left (-(t-4)^3+e^{3 c_1}\right ){}^3}-4096-1280 e^{3 c_1}+8 e^{6 c_1}}+\frac {i \left (\sqrt {3}+i\right ) (t-4) \left ((t-4)^3+8 e^{3 c_1}\right )}{16 \sqrt [3]{-t^6+24 t^5-240 t^4+1280 t^3+20 e^{3 c_1} t^3-3840 t^2-240 e^{3 c_1} t^2+6144 t+960 e^{3 c_1} t+8 \sqrt {e^{3 c_1} \left (-(t-4)^3+e^{3 c_1}\right ){}^3}-4096-1280 e^{3 c_1}+8 e^{6 c_1}}}-t+7 \\ \end{align*}