57.1.65 problem 65

Internal problem ID [9049]
Book : First order enumerated odes
Section : section 1
Problem number : 65
Date solved : Monday, January 27, 2025 at 05:31:21 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} y^{\prime }&=\left (a +b x +c y\right )^{6} \end{align*}

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 94

dsolve(diff(y(x),x)=(a+b*x+c*y(x))^6,y(x), singsol=all)
 

\[ y = \frac {\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{c^{7} \textit {\_a}^{6}+6 \textit {\_a}^{5} a \,c^{6}+15 \textit {\_a}^{4} a^{2} c^{5}+20 \textit {\_a}^{3} a^{3} c^{4}+15 \textit {\_a}^{2} a^{4} c^{3}+6 \textit {\_a} \,a^{5} c^{2}+a^{6} c +b}d \textit {\_a} \right ) c -x +c_{1} \right ) c -b x}{c} \]

✓ Solution by Mathematica

Time used: 1.772 (sec). Leaf size: 274

DSolve[D[y[x],x]==(a+b*x+c*y[x])^6,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\frac {-4 \sqrt [6]{b} \arctan \left (\frac {\sqrt [6]{c} (a+b x+c y(x))}{\sqrt [6]{b}}\right )+2 \sqrt [6]{b} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} (a+b x+c y(x))}{\sqrt [6]{b}}\right )-2 \sqrt [6]{b} \arctan \left (\frac {2 \sqrt [6]{c} (a+b x+c y(x))}{\sqrt [6]{b}}+\sqrt {3}\right )+\sqrt {3} \sqrt [6]{b} \log \left (\sqrt [3]{c} (a+b x+c y(x))^2-\sqrt {3} \sqrt [6]{b} \sqrt [6]{c} (a+b x+c y(x))+\sqrt [3]{b}\right )-\sqrt {3} \sqrt [6]{b} \log \left (\sqrt [3]{c} (a+b x+c y(x))^2+\sqrt {3} \sqrt [6]{b} \sqrt [6]{c} (a+b x+c y(x))+\sqrt [3]{b}\right )+12 a \sqrt [6]{c}+12 b \sqrt [6]{c} x+12 c^{7/6} y(x)}{12 b \sqrt [6]{c}}-\frac {c y(x)}{b}=c_1,y(x)\right ] \]