83.21.10 problem 10

Internal problem ID [19254]
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 : 10
Date solved : Tuesday, January 28, 2025 at 01:15:44 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} 4 y {y^{\prime }}^{2}+2 x y^{\prime }-y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 65

dsolve(4*y(x)*diff(y(x),x)^2+2*x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -\frac {i x}{2} \\ y \left (x \right ) &= \frac {i x}{2} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sqrt {c_{1} \left (-x +c_{1} \right )} \\ y \left (x \right ) &= \sqrt {c_{1} \left (x +c_{1} \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (-x +c_{1} \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (x +c_{1} \right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.513 (sec). Leaf size: 140

DSolve[4*y[x]*D[y[x],x]^2+2*x*D[y[x],x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {1}{2} e^{2 c_1} \sqrt {-2 x+e^{4 c_1}} \\ y(x)\to \frac {1}{2} e^{2 c_1} \sqrt {-2 x+e^{4 c_1}} \\ y(x)\to -\frac {1}{2} e^{2 c_1} \sqrt {2 x+e^{4 c_1}} \\ y(x)\to \frac {1}{2} e^{2 c_1} \sqrt {2 x+e^{4 c_1}} \\ y(x)\to 0 \\ y(x)\to -\frac {i x}{2} \\ y(x)\to \frac {i x}{2} \\ \end{align*}