60.1.42 problem 42

Internal problem ID [10056]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 42
Date solved : Tuesday, January 28, 2025 at 04:15:32 PM
CAS classification : [_Abel]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 53

dsolve(diff(y(x),x) - x*(x+2)*y(x)^3 - (x+3)*y(x)^2=0,y(x), singsol=all)
 

\[ \frac {\frac {\sqrt {2+\left (x^{2}+2 x \right ) y}}{2}+\left (\operatorname {arctanh}\left (\frac {\sqrt {y}\, x}{\sqrt {2+\left (x^{2}+2 x \right ) y}}\right )+c_{1} \right ) \sqrt {y}}{\sqrt {y}} = 0 \]

✓ Solution by Mathematica

Time used: 0.710 (sec). Leaf size: 485

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

\[ \text {Solve}\left [c_1=-\frac {\frac {i \sqrt {\frac {2}{\pi }} \sqrt {\frac {1}{2 y(x)}+\frac {1}{4} (x+1)^2-\frac {1}{4}} \left (\frac {\sinh \left (\sqrt {\frac {1}{2 y(x)}+\frac {1}{4} (x+1)^2-\frac {1}{4}}\right )}{\sqrt {\frac {1}{2 y(x)}+\frac {1}{4} (x+1)^2-\frac {1}{4}}}-\cosh \left (\sqrt {\frac {1}{2 y(x)}+\frac {1}{4} (x+1)^2-\frac {1}{4}}\right )\right )}{\sqrt {-i \sqrt {\frac {1}{2 y(x)}+\frac {1}{4} (x+1)^2-\frac {1}{4}}}}-\frac {i \sqrt {\frac {2}{\pi }} \left (\frac {x+1}{2}+\frac {1}{2}\right ) \sinh \left (\sqrt {\frac {1}{2 y(x)}+\frac {1}{4} (x+1)^2-\frac {1}{4}}\right )}{\sqrt {-i \sqrt {\frac {1}{2 y(x)}+\frac {1}{4} (x+1)^2-\frac {1}{4}}}}}{\frac {i \sqrt {\frac {2}{\pi }} \sqrt {\frac {1}{2 y(x)}+\frac {1}{4} (x+1)^2-\frac {1}{4}} \left (i \sinh \left (\sqrt {\frac {1}{2 y(x)}+\frac {1}{4} (x+1)^2-\frac {1}{4}}\right )-\frac {i \cosh \left (\sqrt {\frac {1}{2 y(x)}+\frac {1}{4} (x+1)^2-\frac {1}{4}}\right )}{\sqrt {\frac {1}{2 y(x)}+\frac {1}{4} (x+1)^2-\frac {1}{4}}}\right )}{\sqrt {-i \sqrt {\frac {1}{2 y(x)}+\frac {1}{4} (x+1)^2-\frac {1}{4}}}}-\frac {\sqrt {\frac {2}{\pi }} \left (\frac {x+1}{2}+\frac {1}{2}\right ) \cosh \left (\sqrt {\frac {1}{2 y(x)}+\frac {1}{4} (x+1)^2-\frac {1}{4}}\right )}{\sqrt {-i \sqrt {\frac {1}{2 y(x)}+\frac {1}{4} (x+1)^2-\frac {1}{4}}}}},y(x)\right ] \]