40.17.9 problem 19

Internal problem ID [6810]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 26. Integration in series (singular points). Supplemetary problems. Page 218
Problem number : 19
Date solved : Monday, January 27, 2025 at 02:30:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x y^{\prime \prime }+y^{\prime }-y&=1+x \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 60

Order:=6; 
dsolve(2*x*diff(y(x),x$2)+diff(y(x),x)-y(x)=x+1,y(x),type='series',x=0);
 

\[ y = c_1 \sqrt {x}\, \left (1+\frac {1}{3} x +\frac {1}{30} x^{2}+\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}+\frac {1}{1247400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+x +\frac {1}{6} x^{2}+\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}+\frac {1}{113400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x \left (1+\frac {1}{3} x +\frac {1}{45} x^{2}+\frac {1}{1260} x^{3}+\frac {1}{56700} x^{4}+\operatorname {O}\left (x^{5}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 246

AsymptoticDSolveValue[2*x*D[y[x],{x,2}]+D[y[x],x]-y[x]==x+1,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {x^5}{113400}+\frac {x^4}{2520}+\frac {x^3}{90}+\frac {x^2}{6}+x+1\right )+c_2 \sqrt {x} \left (\frac {x^5}{1247400}+\frac {x^4}{22680}+\frac {x^3}{630}+\frac {x^2}{30}+\frac {x}{3}+1\right )+\sqrt {x} \left (\frac {x^5}{1247400}+\frac {x^4}{22680}+\frac {x^3}{630}+\frac {x^2}{30}+\frac {x}{3}+1\right ) \left (\frac {23 x^{11/2}}{311850}+\frac {29 x^{9/2}}{11340}+\frac {16 x^{7/2}}{315}+\frac {7 x^{5/2}}{15}+\frac {4 x^{3/2}}{3}+2 \sqrt {x}\right )+\left (\frac {x^5}{113400}+\frac {x^4}{2520}+\frac {x^3}{90}+\frac {x^2}{6}+x+1\right ) \left (-\frac {x^6}{133650}-\frac {37 x^5}{113400}-\frac {11 x^4}{1260}-\frac {11 x^3}{90}-\frac {2 x^2}{3}-x\right ) \]