8.3.13 problem 14

Internal problem ID [689]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.4. Separable equations. Page 43
Problem number : 14
Date solved : Monday, January 27, 2025 at 02:58:06 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 21

dsolve(diff(y(x),x) = (1+x^(1/2))/(1+y(x)^(1/2)),y(x), singsol=all)
 

\[ x +\frac {2 x^{{3}/{2}}}{3}-y-\frac {2 y^{{3}/{2}}}{3}+c_1 = 0 \]

✓ Solution by Mathematica

Time used: 4.078 (sec). Leaf size: 796

DSolve[D[y[x],x]== (1+x^(1/2))/(1+y[x]^(1/2)),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {-16 x^{3/2}+\left (96 x^{5/2}+24 (-3+4 c_1) x^{3/2}+8 \sqrt {\left (2 x^{3/2}+3 x-1+3 c_1\right ) \left (2 x^{3/2}+3 x+3 c_1\right ){}^3}+32 x^3+72 x^2+36 (-3+4 c_1) x+27+72 c_1{}^2-108 c_1\right ){}^{2/3}+3 \sqrt [3]{96 x^{5/2}+24 (-3+4 c_1) x^{3/2}+8 \sqrt {\left (2 x^{3/2}+3 x-1+3 c_1\right ) \left (2 x^{3/2}+3 x+3 c_1\right ){}^3}+32 x^3+72 x^2+36 (-3+4 c_1) x+27+72 c_1{}^2-108 c_1}-24 x+9-24 c_1}{4 \sqrt [3]{96 x^{5/2}+24 (-3+4 c_1) x^{3/2}+8 \sqrt {\left (2 x^{3/2}+3 x-1+3 c_1\right ) \left (2 x^{3/2}+3 x+3 c_1\right ){}^3}+32 x^3+72 x^2+36 (-3+4 c_1) x+27+72 c_1{}^2-108 c_1}} \\ y(x)\to \frac {1}{16} \left (\frac {2 \left (1+i \sqrt {3}\right ) \left (16 x^{3/2}+24 x-9+24 c_1\right )}{\sqrt [3]{96 x^{5/2}+24 (-3+4 c_1) x^{3/2}+8 \sqrt {\left (2 x^{3/2}+3 x-1+3 c_1\right ) \left (2 x^{3/2}+3 x+3 c_1\right ){}^3}+32 x^3+72 x^2+36 (-3+4 c_1) x+27+72 c_1{}^2-108 c_1}}+2 i \left (\sqrt {3}+i\right ) \sqrt [3]{96 x^{5/2}+24 (-3+4 c_1) x^{3/2}+8 \sqrt {\left (2 x^{3/2}+3 x-1+3 c_1\right ) \left (2 x^{3/2}+3 x+3 c_1\right ){}^3}+32 x^3+72 x^2+36 (-3+4 c_1) x+27+72 c_1{}^2-108 c_1}+12\right ) \\ y(x)\to \frac {1}{16} \left (\frac {2 \left (1-i \sqrt {3}\right ) \left (16 x^{3/2}+24 x-9+24 c_1\right )}{\sqrt [3]{96 x^{5/2}+24 (-3+4 c_1) x^{3/2}+8 \sqrt {\left (2 x^{3/2}+3 x-1+3 c_1\right ) \left (2 x^{3/2}+3 x+3 c_1\right ){}^3}+32 x^3+72 x^2+36 (-3+4 c_1) x+27+72 c_1{}^2-108 c_1}}-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{96 x^{5/2}+24 (-3+4 c_1) x^{3/2}+8 \sqrt {\left (2 x^{3/2}+3 x-1+3 c_1\right ) \left (2 x^{3/2}+3 x+3 c_1\right ){}^3}+32 x^3+72 x^2+36 (-3+4 c_1) x+27+72 c_1{}^2-108 c_1}+12\right ) \\ \end{align*}