60.2.196 problem 772

Internal problem ID [10783]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 772
Date solved : Tuesday, January 28, 2025 at 05:13:35 PM
CAS classification : [`x=_G(y,y')`]

\begin{align*} y^{\prime }&=\frac {\left (x +1+\ln \left (y\right ) x \right ) \ln \left (y\right ) y}{x \left (1+x \right )} \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 18

dsolve(diff(y(x),x) = (x+1+ln(y(x))*x)*ln(y(x))*y(x)/x/(x+1),y(x), singsol=all)
 

\[ y = {\mathrm e}^{\frac {x}{\ln \left (x +1\right )+c_{1} -x}} \]

✓ Solution by Mathematica

Time used: 0.155 (sec). Leaf size: 91

DSolve[D[y[x],x] == (Log[y[x]]*(1 + x + x*Log[y[x]])*y[x])/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {x}{K[2] \log ^2(K[2])}-\int _1^x\left (\frac {1}{K[2] \log (K[2])}-\frac {\log (K[2])+1}{K[2] \log ^2(K[2])}\right )dK[1]\right )dK[2]+\int _1^x\left (\frac {\log (y(x))+1}{\log (y(x))}-\frac {1}{K[1]+1}\right )dK[1]=c_1,y(x)\right ] \]