83.35.12 problem 12

Internal problem ID [19432]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Misc. Exercise on chapter VII. Page 118
Problem number : 12
Date solved : Tuesday, January 28, 2025 at 01:42:16 PM
CAS classification : [[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]

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

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 61

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

\begin{align*} y \left (x \right ) &= \frac {4 \left (a^{2}+c_{1} x \right )^{{5}/{2}}+15 c_{1}^{2} \left (c_{2} x +c_3 \right )}{15 c_{1}^{2}} \\ y \left (x \right ) &= \frac {-4 \left (a^{2}+c_{1} x \right )^{{5}/{2}}+15 c_{1}^{2} \left (c_{2} x +c_3 \right )}{15 c_{1}^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.872 (sec). Leaf size: 75

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

\begin{align*} y(x)\to -\frac {4}{15} e^{-4 c_1} \left (a^2+e^{2 c_1} x\right ){}^{5/2}+c_3 x+c_2 \\ y(x)\to \frac {4}{15} e^{-4 c_1} \left (a^2+e^{2 c_1} x\right ){}^{5/2}+c_3 x+c_2 \\ \end{align*}