56.4.18 problem 18
Internal
problem
ID
[8907]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
4.0
Problem
number
:
18
Date
solved
:
Monday, January 27, 2025 at 05:19:05 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 2 x^{2} y^{\prime \prime }+3 x y^{\prime }-y x&=x^{2}+2 x \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Solution by Maple
Time used: 0.016 (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);
\[
y = \frac {c_{1} \left (1+x +\frac {1}{6} x^{2}+\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}+\frac {1}{113400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} \left (1+\frac {1}{3} x +\frac {1}{30} x^{2}+\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}+\frac {1}{1247400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x \left (\frac {2}{3}+\frac {1}{6} x +\frac {1}{126} x^{2}+\frac {1}{4536} x^{3}+\frac {1}{249480} x^{4}+\operatorname {O}\left (x^{5}\right )\right )
\]
✓ Solution by Mathematica
Time used: 0.056 (sec). Leaf size: 239
AsymptoticDSolveValue[2*x^2*D[y[x],{x,2}]+3*x*D[y[x],x]-x*y[x]==x^2+2*x,y[x],{x,0,"6"-1}]
\[
y(x)\to c_1 \left (\frac {x^5}{1247400}+\frac {x^4}{22680}+\frac {x^3}{630}+\frac {x^2}{30}+\frac {x}{3}+1\right )+\frac {c_2 \left (\frac {x^5}{113400}+\frac {x^4}{2520}+\frac {x^3}{90}+\frac {x^2}{6}+x+1\right )}{\sqrt {x}}+\frac {\left (\frac {x^5}{113400}+\frac {x^4}{2520}+\frac {x^3}{90}+\frac {x^2}{6}+x+1\right ) \left (-\frac {19 x^{11/2}}{62370}-\frac {23 x^{9/2}}{2835}-\frac {4 x^{7/2}}{35}-\frac {2 x^{5/2}}{3}-\frac {4 x^{3/2}}{3}\right )}{\sqrt {x}}+\left (\frac {x^5}{1247400}+\frac {x^4}{22680}+\frac {x^3}{630}+\frac {x^2}{30}+\frac {x}{3}+1\right ) \left (\frac {47 x^6}{680400}+\frac {x^5}{420}+\frac {17 x^4}{360}+\frac {4 x^3}{9}+\frac {3 x^2}{2}+2 x\right )
\]