71.9.4 problem 4

Internal problem ID [14483]
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 : 4
Date solved : Tuesday, January 28, 2025 at 06:42:12 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 (5\right )&=0\\ y^{\prime }\left (5\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.055 (sec). Leaf size: 24

dsolve([x*(x-3)*diff(y(x),x$2)+3*diff(y(x),x)=x^2,y(5) = 0, D(y)(5) = 1],y(x), singsol=all)
 

\[ y = \frac {x^{2}}{2}-\frac {8 x}{5}-\frac {24 \ln \left (x -3\right )}{5}+\frac {24 \ln \left (2\right )}{5}-\frac {9}{2} \]

✓ Solution by Mathematica

Time used: 3.746 (sec). Leaf size: 325

DSolve[{x*(x-3)*D[y[x],{x,2}]+3*D[y[x],x]==x^2,{y[5]==0,Derivative[1][y][5]==1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \int _1^x\exp \left (\int _1^{K[3]}-\frac {3}{(K[1]-3) K[1]}dK[1]-\int _1^5-\frac {3}{(K[1]-3) K[1]}dK[1]\right ) \left (-\exp \left (\int _1^5-\frac {3}{(K[1]-3) K[1]}dK[1]\right ) \int _1^5\frac {\exp \left (-\int _1^{K[2]}-\frac {3}{(K[1]-3) K[1]}dK[1]\right ) K[2]}{K[2]-3}dK[2]+\exp \left (\int _1^5-\frac {3}{(K[1]-3) K[1]}dK[1]\right ) \int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}-\frac {3}{(K[1]-3) K[1]}dK[1]\right ) K[2]}{K[2]-3}dK[2]+1\right )dK[3]-\int _1^5\exp \left (\int _1^{K[3]}-\frac {3}{(K[1]-3) K[1]}dK[1]\right ) \left (-\int _1^5\frac {\exp \left (-\int _1^{K[2]}-\frac {3}{(K[1]-3) K[1]}dK[1]\right ) K[2]}{K[2]-3}dK[2]+\exp \left (-\int _1^5-\frac {3}{(K[1]-3) K[1]}dK[1]\right )+\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}-\frac {3}{(K[1]-3) K[1]}dK[1]\right ) K[2]}{K[2]-3}dK[2]\right )dK[3] \]