82.18.4 problem Ex. 4

Internal problem ID [18826]
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. 4
Date solved : Tuesday, January 28, 2025 at 12:22:59 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 \left (x \right ) &= -\frac {1}{4 x^{2}} \\ y \left (x \right ) &= \frac {-i c_{1} -x}{c_{1}^{2} x} \\ y \left (x \right ) &= \frac {i c_{1} -x}{x \,c_{1}^{2}} \\ y \left (x \right ) &= \frac {i c_{1} -x}{x \,c_{1}^{2}} \\ y \left (x \right ) &= \frac {-i c_{1} -x}{c_{1}^{2} x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.527 (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*}