6.3 problem Exercise 12.3, page 103

Internal problem ID [4016]

Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number: Exercise 12.3, page 103.
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: 0.016 (sec). Leaf size: 160

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

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

Solution by Mathematica

Time used: 0.409 (sec). Leaf size: 60

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

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