35.7.18 problem 21

Internal problem ID [6200]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 7. Other second-Order equations. page 435
Problem number : 21
Date solved : Monday, January 27, 2025 at 01:47:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+y&=3 x^{2} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

Time used: 0.550 (sec). Leaf size: 173

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

\[ y(x)\to \frac {1}{2} x^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} \left (\left (\left (1+i \sqrt {3}\right ) x^{\frac {3}{2}+i \sqrt {3}}+\left (1-i \sqrt {3}\right ) x^{3/2}+2 c_1 x^{\frac {i \sqrt {3}}{2}}\right ) \cos \left (\frac {1}{2} \sqrt {3} \log (x)\right )+\left (\left (\sqrt {3}-i\right ) x^{\frac {3}{2}+i \sqrt {3}}+\left (\sqrt {3}+i\right ) x^{3/2}+2 c_2 x^{\frac {i \sqrt {3}}{2}}\right ) \sin \left (\frac {1}{2} \sqrt {3} \log (x)\right )\right ) \]