83.18.9 problem 9

Internal problem ID [19219]
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 (A) at page 53
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 01:14:02 PM
CAS classification : [_quadrature]

\begin{align*} x +y {y^{\prime }}^{2}&=y^{\prime } \left (1+y x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 33

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

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

✓ Solution by Mathematica

Time used: 0.083 (sec). Leaf size: 52

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

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