60.1.73 problem 73

Internal problem ID [10087]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 73
Date solved : Monday, January 27, 2025 at 06:25:53 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }-\left (\frac {a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{a_{3} y^{3}+a_{2} y^{2}+a_{1} y+a_{0}}\right )^{{2}/{3}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 91

dsolve(diff(y(x),x) - ((a__3*x^3+a__2*x^2+a__1*x+a__0)/(a__3*y(x)^3+a__2*y(x)^2+a__1*y(x)+a__0))^(2/3)=0,y(x), singsol=all)
 

\[ \int _{}^{y}\left (\textit {\_a}^{3} a_{3} +\textit {\_a}^{2} a_{2} +\textit {\_a} a_{1} +a_{0} \right )^{{2}/{3}}d \textit {\_a} -\left (a_{3} y^{3}+a_{2} y^{2}+a_{1} y+a_{0} \right )^{{2}/{3}} \left (\int _{}^{x}\left (\frac {\textit {\_a}^{3} a_{3} +\textit {\_a}^{2} a_{2} +\textit {\_a} a_{1} +a_{0}}{a_{3} y^{3}+a_{2} y^{2}+a_{1} y+a_{0}}\right )^{{2}/{3}}d \textit {\_a} \right )+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 21.123 (sec). Leaf size: 733

DSolve[D[y[x],x] - ((a3*x^3+a2*x^2+a1*x+a0)/(a3*y[x]^3+a2*y[x]^2+a1*y[x]+a0))^(2/3)==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\frac {3 (\text {a0}+y(x) (\text {a1}+y(x) (\text {a2}+\text {a3} y(x))))^{2/3} \left (y(x)-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,1\right ]\right ) \operatorname {AppellF1}\left (\frac {5}{3},-\frac {2}{3},-\frac {2}{3},\frac {8}{3},\frac {\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\&,1\right ]-y(x)}{\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\&,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\&,2\right ]},\frac {\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\&,1\right ]-y(x)}{\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\&,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\&,3\right ]}\right )}{5 \left (\frac {y(x)-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,2\right ]}{\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,1\right ]-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,2\right ]}\right )^{2/3} \left (\frac {y(x)-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,3\right ]}{\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,1\right ]-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,3\right ]}\right )^{2/3}}=\frac {3 (\text {a0}+x (\text {a1}+x (\text {a2}+\text {a3} x)))^{2/3} \left (x-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,1\right ]\right ) \operatorname {AppellF1}\left (\frac {5}{3},-\frac {2}{3},-\frac {2}{3},\frac {8}{3},\frac {\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\&,1\right ]-x}{\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\&,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\&,2\right ]},\frac {\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\&,1\right ]-x}{\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\&,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\&,3\right ]}\right )}{5 \left (\frac {x-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,2\right ]}{\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,1\right ]-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,2\right ]}\right )^{2/3} \left (\frac {x-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,3\right ]}{\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,1\right ]-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,3\right ]}\right )^{2/3}}+c_1,y(x)\right ] \]