67.2.13 problem Problem 1(m)
Internal
problem
ID
[13970]
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
:
Tuesday, January 28, 2025 at 08:25:04 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} 2 y^{\prime \prime }+3 y^{\prime }+4 x^{2} y&=1 \end{align*}
✓ Solution by Maple
Time used: 0.289 (sec). Leaf size: 323
dsolve(2*diff(y(x),x$2)+3*diff(y(x),x)+4*x^2*y(x)=1,y(x), singsol=all)
\[
y = x \,{\mathrm e}^{-\frac {x \left (i \sqrt {2}\, x +\frac {3}{2}\right )}{2}} \left (32 \operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \left (\int \frac {\operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) {\mathrm e}^{\frac {i \sqrt {2}\, x^{2}}{2}+\frac {3 x}{4}}}{\left (9 i \sqrt {2}+96\right ) \operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \operatorname {KummerM}\left (-\frac {9 i \sqrt {2}}{128}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )+128 \operatorname {KummerU}\left (-\frac {9 i \sqrt {2}}{128}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )}d x \right )+\operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) c_{2} -32 \left (\int \frac {{\mathrm e}^{\frac {i \sqrt {2}\, x^{2}}{2}+\frac {3 x}{4}} \operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )}{\left (9 i \sqrt {2}+96\right ) \operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \operatorname {KummerM}\left (-\frac {9 i \sqrt {2}}{128}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )+128 \operatorname {KummerU}\left (-\frac {9 i \sqrt {2}}{128}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )}d x \right ) \operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )+\operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) c_{1} \right )
\]
✓ Solution by Mathematica
Time used: 6.483 (sec). Leaf size: 553
DSolve[2*D[y[x],{x,2}]+3*D[y[x],x]+4*x^2*y[x]==1,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to e^{\frac {1}{4} x \left (-3-2 i \sqrt {2} x\right )} \left (\operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {9 i}{64 \sqrt {2}},\frac {1}{2},i \sqrt {2} x^2\right ) \int _1^x\frac {(8+8 i) e^{\frac {1}{4} K[2] \left (2 i \sqrt {2} K[2]+3\right )} \operatorname {HermiteH}\left (-\frac {1}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} K[2]\right )}{\left (9+16 i \sqrt {2}\right ) \left (\sqrt [4]{2} \operatorname {HermiteH}\left (-\frac {3}{2}+\frac {9 i}{32 \sqrt {2}},\frac {(1+i) K[2]}{\sqrt [4]{2}}\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {9 i}{64 \sqrt {2}},\frac {1}{2},i \sqrt {2} K[2]^2\right )+(1+i) \operatorname {HermiteH}\left (-\frac {1}{2}+\frac {9 i}{32 \sqrt {2}},\frac {(1+i) K[2]}{\sqrt [4]{2}}\right ) \operatorname {Hypergeometric1F1}\left (\frac {5}{4}-\frac {9 i}{64 \sqrt {2}},\frac {3}{2},i \sqrt {2} K[2]^2\right ) K[2]\right )}dK[2]+\operatorname {HermiteH}\left (-\frac {1}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} x\right ) \int _1^x\frac {16 e^{\frac {1}{4} K[1] \left (2 i \sqrt {2} K[1]+3\right )} \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {9 i}{64 \sqrt {2}},\frac {1}{2},i \sqrt {2} K[1]^2\right )}{\sqrt [4]{-2} \left (-32+9 i \sqrt {2}\right ) \operatorname {HermiteH}\left (-\frac {3}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} K[1]\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {9 i}{64 \sqrt {2}},\frac {1}{2},i \sqrt {2} K[1]^2\right )+2 \left (-9-16 i \sqrt {2}\right ) \operatorname {HermiteH}\left (-\frac {1}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} K[1]\right ) \operatorname {Hypergeometric1F1}\left (\frac {5}{4}-\frac {9 i}{64 \sqrt {2}},\frac {3}{2},i \sqrt {2} K[1]^2\right ) K[1]}dK[1]+c_1 \operatorname {HermiteH}\left (-\frac {1}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} x\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {9 i}{64 \sqrt {2}},\frac {1}{2},i \sqrt {2} x^2\right )\right )
\]