83.32.2 problem 2

Internal problem ID [19401]
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. Exercise VII (F) at page 113
Problem number : 2
Date solved : Tuesday, January 28, 2025 at 01:37:55 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]]

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

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 71

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

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

✓ Solution by Mathematica

Time used: 26.957 (sec). Leaf size: 106

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

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