71.9.3 problem 3

Internal problem ID [14482]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 4. N-th Order Linear Differential Equations. Exercises 4.1, page 186
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 06:42:10 AM
CAS classification : [[_2nd_order, _missing_y]]

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

With initial conditions

\begin{align*} y \left (1\right )&=0\\ y^{\prime }\left (1\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 11

dsolve([x*(x-3)*diff(y(x),x$2)+3*diff(y(x),x)=x^2,y(1) = 0, D(y)(1) = 1],y(x), singsol=all)
 

\[ y = \frac {x^{2}}{2}-\frac {1}{2} \]

✓ Solution by Mathematica

Time used: 3.703 (sec). Leaf size: 78

DSolve[{x*(x-3)*D[y[x],{x,2}]+3*D[y[x],x]==x^2,{y[1]==0,Derivative[1][y][1]==1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \int _1^x\exp \left (\int _1^{K[3]}-\frac {3}{(K[1]-3) K[1]}dK[1]\right ) \left (\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}-\frac {3}{(K[1]-3) K[1]}dK[1]\right ) K[2]}{K[2]-3}dK[2]+1\right )dK[3] \]