15.19.18 problem 18

Internal problem ID [3302]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 37, page 171
Problem number : 18
Date solved : Monday, January 27, 2025 at 07:31:50 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.224 (sec). Leaf size: 21

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

\begin{align*} y \left (x \right ) &= \frac {x^{4}}{8} \\ y \left (x \right ) &= c_{1} \left (x^{2}-2 c_{1} \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.936 (sec). Leaf size: 142

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

\begin{align*} \text {Solve}\left [\frac {1}{4} \log (y(x))-\frac {\sqrt {x^6-8 x^2 y(x)} \text {arctanh}\left (\frac {\sqrt {x^4-8 y(x)}}{x^2}\right )}{2 x \sqrt {x^4-8 y(x)}}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\sqrt {x^6-8 x^2 y(x)} \text {arctanh}\left (\frac {\sqrt {x^4-8 y(x)}}{x^2}\right )}{2 x \sqrt {x^4-8 y(x)}}+\frac {1}{4} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to \frac {x^4}{8} \\ \end{align*}