62.7.5 problem Ex 5

Internal problem ID [12828]
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
Date solved : Tuesday, January 28, 2025 at 04:25:42 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} y^{\prime }-\frac {y+1}{1+x}&=\sqrt {y+1} \end{align*}

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

Time used: 0.514 (sec). Leaf size: 214

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\int _1^x\left (\frac {2 (K[1]+1)}{\left (K[1]^2+2 K[1]-K[2]\right )^2}+\frac {2 \sqrt {K[2]+1}}{\left (K[1]^2+2 K[1]-K[2]\right )^2}+\frac {1}{\left (K[1]^2+2 K[1]-K[2]\right ) \sqrt {K[2]+1}}\right )dK[1]+\frac {1}{(-x-1) \sqrt {K[2]+1}}+\frac {\sqrt {K[2]+1}}{(x+1) \left (-x^2-2 x+K[2]\right )}+\frac {1}{-x^2-2 x+K[2]}\right )dK[2]+\int _1^x\left (\frac {2 (K[1]+1)}{K[1]^2+2 K[1]-y(x)}+\frac {2 \sqrt {y(x)+1}}{K[1]^2+2 K[1]-y(x)}-\frac {1}{K[1]+1}\right )dK[1]=c_1,y(x)\right ] \]