44.5.75 problem 64

Internal problem ID [7137]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.2 Separable equations. Exercises 2.2 at page 53
Problem number : 64
Date solved : Tuesday, February 04, 2025 at 12:33:15 AM
CAS classification : [_quadrature]

\begin{align*} 1+{x^{\prime }}^{2}&=\frac {a}{y} \end{align*}

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 71

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

\begin{align*} x &= \sqrt {y \left (a -y \right )}-\frac {a \arctan \left (\frac {a -2 y}{2 \sqrt {y \left (a -y \right )}}\right )}{2}+c_1 \\ x &= -\sqrt {y \left (a -y \right )}+\frac {a \arctan \left (\frac {a -2 y}{2 \sqrt {y \left (a -y \right )}}\right )}{2}+c_1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.174 (sec). Leaf size: 314

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

\begin{align*} x(y)\to -\frac {\left (\sqrt {y}-1\right ) \left (\sqrt {a-1}-\sqrt {a-y}\right ) \left (-a^2+a \left (\sqrt {a-1} \sqrt {a-y}+y+\sqrt {y}+1\right )-2 \left (\sqrt {a-1} \sqrt {y} \sqrt {a-y}+y\right )\right )}{\left (\sqrt {a-1} \sqrt {a-y}-a+\sqrt {y}\right )^2}-2 a \arctan \left (\frac {1-\sqrt {y}}{\sqrt {a-1}-\sqrt {a-y}}\right )+c_1 \\ x(y)\to \frac {\left (\sqrt {y}-1\right ) \left (\sqrt {a-1}-\sqrt {a-y}\right ) \left (-a^2+a \left (\sqrt {a-1} \sqrt {a-y}+y+\sqrt {y}+1\right )-2 \left (\sqrt {a-1} \sqrt {y} \sqrt {a-y}+y\right )\right )}{\left (\sqrt {a-1} \sqrt {a-y}-a+\sqrt {y}\right )^2}+2 a \arctan \left (\frac {1-\sqrt {y}}{\sqrt {a-1}-\sqrt {a-y}}\right )+c_1 \\ \end{align*}