60.1.163 problem 164

Internal problem ID [10177]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 164
Date solved : Monday, January 27, 2025 at 06:31:24 PM
CAS classification : [_rational, _Riccati]

\begin{align*} 2 x^{2} y^{\prime }-2 y^{2}-3 y x +2 a^{2} x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 102

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

\[ y = \frac {\left (-2 c_{1} \sqrt {-\frac {a^{2}}{x}}\, x -x \right ) \sin \left (2 \sqrt {-\frac {a^{2}}{x}}\right )-x \left (c_{1} -2 \sqrt {-\frac {a^{2}}{x}}\right ) \cos \left (2 \sqrt {-\frac {a^{2}}{x}}\right )}{2 \cos \left (2 \sqrt {-\frac {a^{2}}{x}}\right ) c_{1} +2 \sin \left (2 \sqrt {-\frac {a^{2}}{x}}\right )} \]

✓ Solution by Mathematica

Time used: 0.287 (sec). Leaf size: 94

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

\begin{align*} y(x)\to \frac {4 a^2 c_1 \sqrt {x}+2 a \sqrt {x} e^{\frac {4 a}{\sqrt {x}}}-x e^{\frac {4 a}{\sqrt {x}}}+2 a c_1 x}{2 e^{\frac {4 a}{\sqrt {x}}}-4 a c_1} \\ y(x)\to a \left (-\sqrt {x}\right )-\frac {x}{2} \\ \end{align*}