Internal problem ID [10886]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {2 y^{\prime \prime }+3 y^{\prime }+4 x^{2} y-1=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 385
dsolve(2*diff(y(x),x$2)+3*diff(y(x),x)+4*x^2*y(x)=1,y(x), singsol=all)
\[ y \left (x \right ) = x \operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) {\mathrm e}^{-\frac {x \left (i \sqrt {2}\, x +\frac {3}{2}\right )}{2}} c_{2} +x \operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) {\mathrm e}^{-\frac {x \left (i \sqrt {2}\, x +\frac {3}{2}\right )}{2}} c_{1} -32 x \left (\operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \left (\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 \right )-\left (\int \frac {{\mathrm e}^{\frac {i \sqrt {2}\, x^{2}}{2}+\frac {3 x}{4}} \operatorname {KummerU}\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 \right ) \operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )\right ) {\mathrm e}^{-\frac {x \left (i \sqrt {2}\, x +\frac {3}{2}\right )}{2}} \]
✓ Solution by Mathematica
Time used: 3.699 (sec). Leaf size: 547
DSolve[2*y''[x]+3*y'[x]+4*x^2*y[x]==1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} 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 i+16 \sqrt {2}\right ) \left (\sqrt [4]{2} \operatorname {HermiteH}\left (-\frac {3}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} K[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}},\sqrt [4]{-2} K[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 ) \\ \end{align*}