82.54.9 problem Ex. 9

Internal problem ID [19025]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IX. Equations of the second order. problems at end of chapter at page 120
Problem number : Ex. 9
Date solved : Tuesday, January 28, 2025 at 12:46:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }-y&=x \left (-x^{2}+1\right )^{{3}/{2}} \end{align*}

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 49

dsolve((1-x^2)*diff(y(x),x$2)+x*diff(y(x),x)-y(x)=x*(1-x^2)^(3/2),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\sqrt {-x^{2}+1}\, \left (x^{3}-x \right )}{9}+\left (\ln \left (x +\sqrt {x^{2}-1}\right ) c_{1} +c_{2} \right ) x -\sqrt {x^{2}-1}\, c_{1} \]

✓ Solution by Mathematica

Time used: 0.352 (sec). Leaf size: 173

DSolve[(1-x^2)*D[y[x],{x,2}]+x*D[y[x],x]-y[x]==x*(1-x^2)^(3/2),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\left (x^2-1\right ) \left (x^5-2 x^3-9 c_1 \sqrt {1-x^2} x+9 c_2 \sqrt {-\left (x^2-1\right )^2}+x\right )-3 x \text {arctanh}\left (\frac {x}{\sqrt {x^2-1}}\right ) \left (3 c_2 \sqrt {1-x^2} x^2-3 c_2 \sqrt {1-x^2}+\sqrt {x^2-1} x^5-\left (\sqrt {x^2-1}-\sqrt {1-x^2} \sqrt {-\left (x^2-1\right )^2}\right ) x^3\right )}{9 \left (1-x^2\right )^{3/2}} \]