Internal problem ID [6486]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 18.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 y^{\prime \prime } x^{2}+3 y^{\prime } x -x y-x^{2}-2 x=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 60
Order:=6; dsolve(2*x^2*diff(y(x),x$2)+3*x*diff(y(x),x)-x*y(x)=x^2+2*x,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} \left (1+\frac {1}{3} x +\frac {1}{30} x^{2}+\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}+\frac {1}{1247400} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \sqrt {x}+x^{\frac {3}{2}} \left (\frac {2}{3}+\frac {1}{6} x +\frac {1}{126} x^{2}+\frac {1}{4536} x^{3}+\frac {1}{249480} x^{4}+\mathrm {O}\left (x^{5}\right )\right )+c_{1} \left (1+x +\frac {1}{6} x^{2}+\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}+\frac {1}{113400} x^{5}+\mathrm {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.078 (sec). Leaf size: 239
AsymptoticDSolveValue[2*x^2*y''[x]+3*x*y'[x]-x*y[x]==x^2+2*x,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {x^5}{1247400}+\frac {x^4}{22680}+\frac {x^3}{630}+\frac {x^2}{30}+\frac {x}{3}+1\right )+\frac {c_2 \left (\frac {x^5}{113400}+\frac {x^4}{2520}+\frac {x^3}{90}+\frac {x^2}{6}+x+1\right )}{\sqrt {x}}+\frac {\left (\frac {x^5}{113400}+\frac {x^4}{2520}+\frac {x^3}{90}+\frac {x^2}{6}+x+1\right ) \left (-\frac {19 x^{11/2}}{62370}-\frac {23 x^{9/2}}{2835}-\frac {4 x^{7/2}}{35}-\frac {2 x^{5/2}}{3}-\frac {4 x^{3/2}}{3}\right )}{\sqrt {x}}+\left (\frac {x^5}{1247400}+\frac {x^4}{22680}+\frac {x^3}{630}+\frac {x^2}{30}+\frac {x}{3}+1\right ) \left (\frac {47 x^6}{680400}+\frac {x^5}{420}+\frac {17 x^4}{360}+\frac {4 x^3}{9}+\frac {3 x^2}{2}+2 x\right ) \]