83.21.6 problem 6

Internal problem ID [19250]
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 (D) at page 57
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 01:15:41 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 37

dsolve((y(x)-diff(y(x),x)*x)*(diff(y(x),x)-1)=diff(y(x),x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 42

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

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