83.22.14 problem 14

Internal problem ID [19268]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 14
Date solved : Tuesday, January 28, 2025 at 01:17:58 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} \left ({y^{\prime }}^{2}-\frac {1}{a^{2}-x^{2}}\right ) \left (y^{\prime }-\sqrt {\frac {y}{x}}\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 75

dsolve((diff(y(x),x)^2- 1/(a^2-x^2))*(diff(y(x),x) - sqrt(y(x)/x))=0,y(x), singsol=all)
 

\begin{align*} -\ln \left (\frac {-x +y \left (x \right )}{x}\right )+2 \,\operatorname {arctanh}\left (\sqrt {\frac {y \left (x \right )}{x}}\right )-\ln \left (x \right )-c_{1} &= 0 \\ y \left (x \right ) &= \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_{1} \\ y \left (x \right ) &= -\arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.088 (sec). Leaf size: 69

DSolve[(D[y[x],x]^2- 1/(a^2-x^2))*(D[y[x],x] - Sqrt[y[x]/x])==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \arctan \left (\frac {x}{\sqrt {a^2-x^2}}\right )+c_1 \\ y(x)\to -\arctan \left (\frac {x}{\sqrt {a^2-x^2}}\right )+c_1 \\ y(x)\to \frac {1}{4} \left (2 \sqrt {x}+c_1\right ){}^2 \\ \end{align*}