73.3.38 problem 4.7 (L)

Internal problem ID [15072]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 4. SEPARABLE FIRST ORDER EQUATIONS. Additional exercises. page 90
Problem number : 4.7 (L)
Date solved : Tuesday, January 28, 2025 at 07:29:40 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {2+\sqrt {x}}{2+\sqrt {y}} \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 23

dsolve(diff(y(x),x)=(2+sqrt(x))/(2+sqrt(y(x))),y(x), singsol=all)
 

\[ 2 x +\frac {2 x^{{3}/{2}}}{3}-2 y-\frac {2 y^{{3}/{2}}}{3}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 4.257 (sec). Leaf size: 1162

DSolve[D[y[x],x]==(2+Sqrt[x])/(2+Sqrt[y[x]]),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {-8 x^{3/2}+\left (24 x^{5/2}+12 (-6+c_1) x^{3/2}+\sqrt {\left (2 x^{3/2}+6 x-8+3 c_1\right ) \left (2 x^{3/2}+6 x+3 c_1\right ){}^3}+4 x^3+36 x^2+36 (-6+c_1) x+216+9 c_1{}^2-108 c_1\right ){}^{2/3}+6 \sqrt [3]{24 x^{5/2}+12 (-6+c_1) x^{3/2}+\sqrt {\left (2 x^{3/2}+6 x-8+3 c_1\right ) \left (2 x^{3/2}+6 x+3 c_1\right ){}^3}+4 x^3+36 x^2+36 (-6+c_1) x+216+9 c_1{}^2-108 c_1}-24 x+36-12 c_1}{2 \sqrt [3]{24 x^{5/2}+12 (-6+c_1) x^{3/2}+\sqrt {\left (2 x^{3/2}+6 x-8+3 c_1\right ) \left (2 x^{3/2}+6 x+3 c_1\right ){}^3}+4 x^3+36 x^2+36 (-6+c_1) x+216+9 c_1{}^2-108 c_1}} \\ y(x)\to \frac {\left (8+8 i \sqrt {3}\right ) x^{3/2}+i \sqrt {3} \left (24 x^{5/2}+12 (-6+c_1) x^{3/2}+\sqrt {\left (2 x^{3/2}+6 x-8+3 c_1\right ) \left (2 x^{3/2}+6 x+3 c_1\right ){}^3}+4 x^3+36 x^2+36 (-6+c_1) x+216+9 c_1{}^2-108 c_1\right ){}^{2/3}-\left (24 x^{5/2}+12 (-6+c_1) x^{3/2}+\sqrt {\left (2 x^{3/2}+6 x-8+3 c_1\right ) \left (2 x^{3/2}+6 x+3 c_1\right ){}^3}+4 x^3+36 x^2+36 (-6+c_1) x+216+9 c_1{}^2-108 c_1\right ){}^{2/3}+12 \sqrt [3]{24 x^{5/2}+12 (-6+c_1) x^{3/2}+\sqrt {\left (2 x^{3/2}+6 x-8+3 c_1\right ) \left (2 x^{3/2}+6 x+3 c_1\right ){}^3}+4 x^3+36 x^2+36 (-6+c_1) x+216+9 c_1{}^2-108 c_1}+24 \left (1+i \sqrt {3}\right ) x-36 i \sqrt {3}-36+12 i \sqrt {3} c_1+12 c_1}{4 \sqrt [3]{24 x^{5/2}+12 (-6+c_1) x^{3/2}+\sqrt {\left (2 x^{3/2}+6 x-8+3 c_1\right ) \left (2 x^{3/2}+6 x+3 c_1\right ){}^3}+4 x^3+36 x^2+36 (-6+c_1) x+216+9 c_1{}^2-108 c_1}} \\ y(x)\to \frac {\left (8-8 i \sqrt {3}\right ) x^{3/2}-i \sqrt {3} \left (24 x^{5/2}+12 (-6+c_1) x^{3/2}+\sqrt {\left (2 x^{3/2}+6 x-8+3 c_1\right ) \left (2 x^{3/2}+6 x+3 c_1\right ){}^3}+4 x^3+36 x^2+36 (-6+c_1) x+216+9 c_1{}^2-108 c_1\right ){}^{2/3}-\left (24 x^{5/2}+12 (-6+c_1) x^{3/2}+\sqrt {\left (2 x^{3/2}+6 x-8+3 c_1\right ) \left (2 x^{3/2}+6 x+3 c_1\right ){}^3}+4 x^3+36 x^2+36 (-6+c_1) x+216+9 c_1{}^2-108 c_1\right ){}^{2/3}+12 \sqrt [3]{24 x^{5/2}+12 (-6+c_1) x^{3/2}+\sqrt {\left (2 x^{3/2}+6 x-8+3 c_1\right ) \left (2 x^{3/2}+6 x+3 c_1\right ){}^3}+4 x^3+36 x^2+36 (-6+c_1) x+216+9 c_1{}^2-108 c_1}+24 \left (1-i \sqrt {3}\right ) x+36 i \sqrt {3}-36-12 i \sqrt {3} c_1+12 c_1}{4 \sqrt [3]{24 x^{5/2}+12 (-6+c_1) x^{3/2}+\sqrt {\left (2 x^{3/2}+6 x-8+3 c_1\right ) \left (2 x^{3/2}+6 x+3 c_1\right ){}^3}+4 x^3+36 x^2+36 (-6+c_1) x+216+9 c_1{}^2-108 c_1}} \\ \end{align*}