82.18.9 problem Ex. 10

Internal problem ID [18831]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Examples on chapter III. page 38
Problem number : Ex. 10
Date solved : Tuesday, January 28, 2025 at 12:24:00 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} 3 y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+4 y^{2}-x^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 105

dsolve(3*diff(y(x),x)^2*y(x)^2-2*x*y(x)*diff(y(x),x)+4*y(x)^2-x^2=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -\frac {\sqrt {3}\, x}{3} \\ y \left (x \right ) &= \frac {\sqrt {3}\, x}{3} \\ \ln \left (x \right )-\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-3 y \left (x \right )^{2}}{x^{2}}}}{2}\right )+\frac {\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right )}{2}-c_{1} &= 0 \\ \ln \left (x \right )+\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-3 y \left (x \right )^{2}}{x^{2}}}}{2}\right )+\frac {\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right )}{2}-c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.763 (sec). Leaf size: 179

DSolve[3*D[y[x],x]^2*y[x]^2-2*x*y[x]*D[y[x],x]+4*y[x]^2-x^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\sqrt {-3 x^2-4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}} \\ y(x)\to \frac {\sqrt {-3 x^2-4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}} \\ y(x)\to -\frac {\sqrt {-3 x^2+4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}} \\ y(x)\to \frac {\sqrt {-3 x^2+4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}} \\ y(x)\to -\sqrt {-x^2} \\ y(x)\to \sqrt {-x^2} \\ \end{align*}