67.2.13 problem Problem 1(m)

Internal problem ID [13899]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number : Problem 1(m)
Date solved : Monday, March 31, 2025 at 08:17:27 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 2 y^{\prime \prime }+3 y^{\prime }+4 x^{2} y&=1 \end{align*}

✓ Maple. Time used: 0.267 (sec). Leaf size: 323

ode:=2*diff(diff(y(x),x),x)+3*diff(y(x),x)+4*x^2*y(x) = 1; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x \,{\mathrm e}^{-\frac {x \left (i \sqrt {2}\, x +\frac {3}{2}\right )}{2}} \left (32 \int \frac {\operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) {\mathrm e}^{\frac {i \sqrt {2}\, x^{2}}{2}+\frac {3 x}{4}}}{\left (9 i \sqrt {2}+96\right ) \operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \operatorname {KummerM}\left (-\frac {9 i \sqrt {2}}{128}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )+128 \operatorname {KummerU}\left (-\frac {9 i \sqrt {2}}{128}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )}d x \operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )-32 \int \frac {{\mathrm e}^{\frac {i \sqrt {2}\, x^{2}}{2}+\frac {3 x}{4}} \operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )}{\left (9 i \sqrt {2}+96\right ) \operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \operatorname {KummerM}\left (-\frac {9 i \sqrt {2}}{128}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )+128 \operatorname {KummerU}\left (-\frac {9 i \sqrt {2}}{128}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )}d x \operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )+\operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) c_1 +\operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) c_2 \right ) \]

✓ Mathematica. Time used: 6.483 (sec). Leaf size: 553

ode=2*D[y[x],{x,2}]+3*D[y[x],x]+4*x^2*y[x]==1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{\frac {1}{4} x \left (-3-2 i \sqrt {2} x\right )} \left (\operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {9 i}{64 \sqrt {2}},\frac {1}{2},i \sqrt {2} x^2\right ) \int _1^x\frac {(8+8 i) e^{\frac {1}{4} K[2] \left (2 i \sqrt {2} K[2]+3\right )} \operatorname {HermiteH}\left (-\frac {1}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} K[2]\right )}{\left (9+16 i \sqrt {2}\right ) \left (\sqrt [4]{2} \operatorname {HermiteH}\left (-\frac {3}{2}+\frac {9 i}{32 \sqrt {2}},\frac {(1+i) K[2]}{\sqrt [4]{2}}\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {9 i}{64 \sqrt {2}},\frac {1}{2},i \sqrt {2} K[2]^2\right )+(1+i) \operatorname {HermiteH}\left (-\frac {1}{2}+\frac {9 i}{32 \sqrt {2}},\frac {(1+i) K[2]}{\sqrt [4]{2}}\right ) \operatorname {Hypergeometric1F1}\left (\frac {5}{4}-\frac {9 i}{64 \sqrt {2}},\frac {3}{2},i \sqrt {2} K[2]^2\right ) K[2]\right )}dK[2]+\operatorname {HermiteH}\left (-\frac {1}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} x\right ) \int _1^x\frac {16 e^{\frac {1}{4} K[1] \left (2 i \sqrt {2} K[1]+3\right )} \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {9 i}{64 \sqrt {2}},\frac {1}{2},i \sqrt {2} K[1]^2\right )}{\sqrt [4]{-2} \left (-32+9 i \sqrt {2}\right ) \operatorname {HermiteH}\left (-\frac {3}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} K[1]\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {9 i}{64 \sqrt {2}},\frac {1}{2},i \sqrt {2} K[1]^2\right )+2 \left (-9-16 i \sqrt {2}\right ) \operatorname {HermiteH}\left (-\frac {1}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} K[1]\right ) \operatorname {Hypergeometric1F1}\left (\frac {5}{4}-\frac {9 i}{64 \sqrt {2}},\frac {3}{2},i \sqrt {2} K[1]^2\right ) K[1]}dK[1]+c_1 \operatorname {HermiteH}\left (-\frac {1}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} x\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {9 i}{64 \sqrt {2}},\frac {1}{2},i \sqrt {2} x^2\right )\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*y(x) + 3*Derivative(y(x), x) + 2*Derivative(y(x), (x, 2)) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE 4*x**2*y(x)/3 + Derivative(y(x), x) + 2*Derivative(y(x), (x, 2))/3 - 1/3 cannot be solved by the factorable group method