40.6.6 problem 15

Internal problem ID [6686]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 10. Singular solutions, Extraneous loci. Supplemetary problems. Page 74
Problem number : 15
Date solved : Monday, January 27, 2025 at 02:22:23 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Solution by Maple

Time used: 0.024 (sec). Leaf size: 77

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

\begin{align*} y &= -\frac {1}{4 x^{2}} \\ y &= \frac {-i c_1 -x}{x \,c_1^{2}} \\ y &= \frac {i c_1 -x}{x \,c_1^{2}} \\ y &= \frac {i c_1 -x}{x \,c_1^{2}} \\ y &= \frac {-i c_1 -x}{x \,c_1^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.535 (sec). Leaf size: 123

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

\begin{align*} \text {Solve}\left [-\frac {x \sqrt {4 x^2 y(x)+1} \text {arctanh}\left (\sqrt {4 x^2 y(x)+1}\right )}{\sqrt {4 x^4 y(x)+x^2}}-\frac {1}{2} \log (y(x))&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {x \sqrt {4 x^2 y(x)+1} \text {arctanh}\left (\sqrt {4 x^2 y(x)+1}\right )}{\sqrt {4 x^4 y(x)+x^2}}-\frac {1}{2} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}