83.48.20 problem Ex 21 page 116

Internal problem ID [19612]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter VII. Exact differential equations.
Problem number : Ex 21 page 116
Date solved : Tuesday, January 28, 2025 at 08:32:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{4} y^{\prime \prime }&=\left (y-x y^{\prime }\right )^{3} \end{align*}

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 37

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

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

✓ Solution by Mathematica

Time used: 60.191 (sec). Leaf size: 95

DSolve[x^4*D[y[x],{x,2}]==(y[x]-x*D[y[x],x])^3,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -i x \log \left (\frac {e^{c_2}-\sqrt {e^{2 c_2}-8 i c_1 x^2}}{4 c_1 x}\right ) \\ y(x)\to -i x \log \left (\frac {\sqrt {e^{2 c_2}-8 i c_1 x^2}+e^{c_2}}{4 c_1 x}\right ) \\ \end{align*}