78.14.16 problem 6 (a)

Internal problem ID [18352]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 19. The Method of Variation of Parameters. Problems at page 135
Problem number : 6 (a)
Date solved : Tuesday, January 28, 2025 at 11:48:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 20

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

\[ y = c_{2} x +c_{1} x^{2}+c_{1} +\frac {1}{2}+\frac {1}{6} x^{4} \]

✓ Solution by Mathematica

Time used: 0.194 (sec). Leaf size: 111

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

\[ y(x)\to \frac {-x^6+4 x^4-2 x^3+3 x^2 \left (-1+2 c_1 \sqrt {-\left (x^2-1\right )^2}\right )+2 x \left (-6 c_1 \sqrt {-\left (x^2-1\right )^2}+3 c_2 \sqrt {-\left (x^2-1\right )^2}+1\right )+6 c_1 \sqrt {-\left (x^2-1\right )^2}}{6-6 x^2} \]