29.8.22 problem 227

Internal problem ID [4827]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 8
Problem number : 227
Date solved : Monday, January 27, 2025 at 09:41:41 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 81

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

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

✓ Solution by Mathematica

Time used: 0.263 (sec). Leaf size: 60

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

\[ \text {Solve}\left [\frac {2 \sqrt {y(x)+1} \arctan \left (\frac {x+1}{\sqrt {-y(x)-1}}\right )}{\sqrt {-y(x)-1}}+\log \left (y(x)-(x+1)^2+1\right )-\log (x+1)=c_1,y(x)\right ] \]