Internal problem ID [2975]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 8
Problem number: 227.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x +1\right ) y^{\prime }-1-y-\left (x +1\right ) \sqrt {1+y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 160
dsolve((1+x)*diff(y(x),x) = 1+y(x)+(1+x)*sqrt(1+y(x)),y(x), singsol=all)
\[ \frac {\sqrt {y \relax (x )+1}\, x}{\left (-x^{2}-2 x +y \relax (x )\right ) \left (\sqrt {y \relax (x )+1}-x -1\right )}+\frac {2 x}{\left (-x^{2}-2 x +y \relax (x )\right ) \left (\sqrt {y \relax (x )+1}-x -1\right )}+\frac {x^{2}}{\left (-x^{2}-2 x +y \relax (x )\right ) \left (\sqrt {y \relax (x )+1}-x -1\right )}+\frac {\sqrt {y \relax (x )+1}}{\left (-x^{2}-2 x +y \relax (x )\right ) \left (\sqrt {y \relax (x )+1}-x -1\right )}+\frac {1}{\left (-x^{2}-2 x +y \relax (x )\right ) \left (\sqrt {y \relax (x )+1}-x -1\right )}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.25 (sec). Leaf size: 60
DSolve[(1+x) y'[x]==(1+y[x])+(1+x)Sqrt[1+y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {2 \sqrt {y(x)+1} \text {ArcTan}\left (\frac {x+1}{\sqrt {-y(x)-1}}\right )}{\sqrt {-y(x)-1}}+\log \left (y(x)-(x+1)^2+1\right )-\log (x+1)=c_1,y(x)\right ] \]