60.1.472 problem 475

Internal problem ID [10486]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 475
Date solved : Monday, January 27, 2025 at 07:55:13 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: 0.855 (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 &= -\frac {i x}{2} \\ y &= \frac {i x}{2} \\ y &= 0 \\ y &= \sqrt {c_{1} \left (-x +c_{1} \right )} \\ y &= \sqrt {c_{1} \left (x +c_{1} \right )} \\ y &= -\sqrt {c_{1} \left (-x +c_{1} \right )} \\ y &= -\sqrt {c_{1} \left (x +c_{1} \right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.523 (sec). Leaf size: 140

DSolve[-y[x] + 2*x*D[y[x],x] + 4*y[x]*D[y[x],x]^2==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*}