Internal problem ID [1267]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN
ORDINARY POINT II. Exercises 7.3. Page 338
Problem number: 29.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (\beta \,x^{2}+x \alpha +1\right ) y^{\prime \prime }+\left (\delta x +\gamma \right ) y^{\prime }+\epsilon y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 366
Order:=6; dsolve((1+alpha*x+beta*x^2)*diff(y(x),x$2)+(gamma+delta*x)*diff(y(x),x)+epsilon*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {\epsilon \,x^{2}}{2}+\frac {\epsilon \left (\alpha +\gamma \right ) x^{3}}{6}+\frac {\epsilon \left (-\alpha ^{2}-\frac {3}{2} \alpha \gamma -\frac {1}{2} \gamma ^{2}+\beta +\delta +\frac {1}{2} \epsilon \right ) x^{4}}{12}-\frac {\epsilon \left (\left (\frac {\alpha }{3}+\frac {\gamma }{6}\right ) \epsilon -\frac {\gamma ^{3}}{12}-\frac {\alpha \,\gamma ^{2}}{2}+\left (-\frac {11 \alpha ^{2}}{12}+\frac {2 \beta }{3}+\frac {5 \delta }{12}\right ) \gamma +\alpha \left (-\frac {\alpha ^{2}}{2}+\beta +\frac {3 \delta }{4}\right )\right ) x^{5}}{10}\right ) y \relax (0)+\left (x -\frac {\gamma \,x^{2}}{2}+\frac {\left (\alpha \gamma +\gamma ^{2}-\delta -\epsilon \right ) x^{3}}{6}+\frac {\left (\left (2 \alpha +2 \gamma \right ) \epsilon -\gamma ^{3}-3 \alpha \,\gamma ^{2}+\left (-2 \alpha ^{2}+2 \beta +3 \delta \right ) \gamma +2 \delta \alpha \right ) x^{4}}{24}+\frac {\left (\epsilon ^{2}+\left (-6 \alpha ^{2}-9 \alpha \gamma -3 \gamma ^{2}+6 \beta +4 \delta \right ) \epsilon +\gamma ^{4}+6 \alpha \,\gamma ^{3}+\left (11 \alpha ^{2}-8 \beta -6 \delta \right ) \gamma ^{2}-12 \left (-\frac {\alpha ^{2}}{2}+\beta +\frac {7 \delta }{6}\right ) \alpha \gamma +6 \left (-\alpha ^{2}+\beta +\frac {\delta }{2}\right ) \delta \right ) x^{5}}{120}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 561
AsymptoticDSolveValue[(1+\[Alpha]*x+\[Beta]*x^2)*y''[x]+(\[Gamma]+\[Delta]*x)*y'[x]+\[Epsilon]*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {1}{20} x^5 \epsilon \left (2 \alpha \beta -\alpha ^3\right )-\frac {1}{20} x^5 \epsilon \left (\alpha \delta -\gamma \left (\alpha ^2-\beta \right )\right )+\frac {1}{60} \gamma x^5 \epsilon \left (\alpha ^2-\beta \right )+\frac {1}{120} \alpha \gamma ^2 x^5 \epsilon +\frac {1}{40} \alpha x^5 \epsilon (\alpha \gamma -\delta )+\frac {1}{24} \gamma x^5 \epsilon (\alpha \gamma -\delta )-\frac {1}{30} \alpha x^5 \epsilon ^2+\frac {1}{120} \gamma ^3 x^5 \epsilon -\frac {1}{60} \gamma x^5 \epsilon ^2-\frac {1}{12} x^4 \epsilon \left (\alpha ^2-\beta \right )-\frac {1}{12} x^4 \epsilon (\alpha \gamma -\delta )-\frac {1}{24} \alpha \gamma x^4 \epsilon -\frac {1}{24} \gamma ^2 x^4 \epsilon +\frac {x^4 \epsilon ^2}{24}+\frac {1}{6} \alpha x^3 \epsilon +\frac {1}{6} \gamma x^3 \epsilon -\frac {x^2 \epsilon }{2}+1\right )+c_2 \left (\frac {1}{60} \gamma x^5 \left (\gamma \left (\alpha ^2-\beta \right )-\alpha \delta \right )-\frac {1}{20} \gamma x^5 \left (\alpha \delta -\gamma \left (\alpha ^2-\beta \right )\right )-\frac {1}{20} x^5 \epsilon \left (\alpha ^2-\beta \right )-\frac {1}{20} x^5 \left (\alpha ^3 (-\gamma )+\alpha ^2 \delta +2 \alpha \beta \gamma -\beta \delta \right )+\frac {1}{24} \gamma ^2 x^5 (\alpha \gamma -\delta )-\frac {1}{120} \gamma ^2 x^5 (\delta -\alpha \gamma )-\frac {1}{40} x^5 \epsilon (\alpha \gamma -\delta )+\frac {1}{120} x^5 \epsilon (\delta -\alpha \gamma )-\frac {1}{40} x^5 (\alpha \gamma -\delta ) (\delta -\alpha \gamma )-\frac {1}{24} \alpha \gamma x^5 \epsilon +\frac {\gamma ^4 x^5}{120}-\frac {1}{40} \gamma ^2 x^5 \epsilon +\frac {x^5 \epsilon ^2}{120}-\frac {1}{12} x^4 \left (\gamma \left (\alpha ^2-\beta \right )-\alpha \delta \right )-\frac {1}{12} \gamma x^4 (\alpha \gamma -\delta )+\frac {1}{24} \gamma x^4 (\delta -\alpha \gamma )+\frac {1}{12} \alpha x^4 \epsilon -\frac {\gamma ^3 x^4}{24}+\frac {1}{12} \gamma x^4 \epsilon -\frac {1}{6} x^3 (\delta -\alpha \gamma )+\frac {\gamma ^2 x^3}{6}-\frac {x^3 \epsilon }{6}-\frac {\gamma x^2}{2}+x\right ) \]