Internal problem ID [10138]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter 2, differential equations of the first order and the first degree. Article 14. Equations
reducible to linear equations (Bernoulli). Page 21
Problem number: Ex 5.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y+1}{x +1}-\sqrt {y+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 160
dsolve(diff(y(x),x)- (y(x)+1)/(x+1)=sqrt(1+y(x)),y(x), singsol=all)
\[ \frac {\sqrt {y \relax (x )+1}}{\left (-x^{2}-2 x +y \relax (x )\right ) \left (\sqrt {y \relax (x )+1}-1-x \right )}+\frac {1}{\left (-x^{2}-2 x +y \relax (x )\right ) \left (\sqrt {y \relax (x )+1}-1-x \right )}+\frac {2 x}{\left (-x^{2}-2 x +y \relax (x )\right ) \left (\sqrt {y \relax (x )+1}-1-x \right )}+\frac {\sqrt {y \relax (x )+1}\, x}{\left (-x^{2}-2 x +y \relax (x )\right ) \left (\sqrt {y \relax (x )+1}-1-x \right )}+\frac {x^{2}}{\left (-x^{2}-2 x +y \relax (x )\right ) \left (\sqrt {y \relax (x )+1}-1-x \right )}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.276 (sec). Leaf size: 60
DSolve[y'[x]- (y[x]+1)/(x+1)==Sqrt[1+y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {2 \sqrt {y(x)+1} \text {ArcTan}\left (\frac {x+1}{\sqrt {-y(x)-1}}\right )}{\sqrt {-y(x)-1}}+\log \left (y(x)-(x+1)^2+1\right )-\log (x+1)=c_1,y(x)\right ] \]