83.45.3 problem Ex 3 page 52

Internal problem ID [19546]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter IV. Equations of the first order but not of the first degree
Problem number : Ex 3 page 52
Date solved : Tuesday, January 28, 2025 at 01:50:32 PM
CAS classification : [_quadrature]

\begin{align*} \left (a^{2}-x^{2}\right ) {y^{\prime }}^{3}+b x \left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}-y^{\prime }-b x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 52

dsolve((a^2-x^2)*diff(y(x),x)^3+b*x*(a^2-x^2)*diff(y(x),x)^2-diff(y(x),x)-b*x=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -\frac {b \,x^{2}}{2}+c_{1} \\ 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: 0.005 (sec). Leaf size: 64

DSolve[(a^2-x^2)*D[y[x],x]^3+b*x*(a^2-x^2)*D[y[x],x]^2-D[y[x],x]-b*x==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {b x^2}{2}+c_1 \\ 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 \\ \end{align*}