83.30.11 problem 11

Internal problem ID [19390]
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 (D) at page 109
Problem number : 11
Date solved : Tuesday, January 28, 2025 at 01:37:09 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime }-x y^{\prime \prime }-\frac {a^{2} y^{\prime }}{x}+\frac {x^{2}}{a}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 55

dsolve(diff(y(x),x)-x*diff(y(x),x$2)-a^2/x*diff(y(x),x)+x^2/a=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {c_{2} a -\int x \left (\operatorname {Ei}_{1}\left (\frac {a^{2}}{x}\right ) {\mathrm e}^{\frac {a^{2}}{x}} a^{2}-{\mathrm e}^{\frac {a^{2}}{x}} c_{1} a -x \right )d x}{a} \]

✓ Solution by Mathematica

Time used: 0.166 (sec). Leaf size: 127

DSolve[D[y[x],x]-x*D[y[x],{x,2}]-a^2/x*D[y[x],x]+x^2/a==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {-6 a^2 x^2 G_{2,3}^{3,1}\left (\frac {a^2}{x}| \begin {array}{c} 0,3 \\ 0,0,2 \\ \end {array} \right )+3 a c_1 x e^{\frac {a^2}{x}} \left (a^2+x\right )+3 a^6 \left (\log \left (-\frac {a^2}{x}\right )-2 \log \left (\frac {a^2}{x}\right )-\log \left (-\frac {x}{a^2}\right )\right ) \Gamma \left (-2,-\frac {a^2}{x}\right )-3 a^5 c_1 \operatorname {ExpIntegralEi}\left (\frac {a^2}{x}\right )+2 x^3}{6 a}+c_2 \]