32.6.3 problem Exercise 12.3, page 103

Internal problem ID [5868]
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
Date solved : Monday, January 27, 2025 at 01:21:45 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 81

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

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

✓ Solution by Mathematica

Time used: 0.258 (sec). Leaf size: 60

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

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