44.1.1 problem 1
Internal
problem
ID
[6876]
Book
:
A
First
Course
in
Differential
Equations
with
Modeling
Applications
by
Dennis
G.
Zill.
12
ed.
Metric
version.
2024.
Cengage
learning.
Section
:
Chapter
1.
Introduction
to
differential
equations.
Exercises
1.1
at
page
12
Problem
number
:
1
Date
solved
:
Tuesday, January 28, 2025 at 03:10:35 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} \left (1-x \right ) y^{\prime \prime }-4 x y^{\prime }+5 y&=\cos \left (x \right ) \end{align*}
✓ Solution by Maple
Time used: 0.038 (sec). Leaf size: 337
dsolve((1-x)*diff(y(x),x$2)-4*x*diff(y(x),x)+5*y(x)=cos(x),y(x), singsol=all)
\[
y = {\mathrm e}^{-4 x} \left (540540 \operatorname {KummerM}\left (\frac {21}{4}, 4, 4 x -4\right ) \left (\int \frac {\cos \left (x \right ) \operatorname {KummerU}\left (\frac {21}{4}, 4, 4 x -4\right ) {\mathrm e}^{4 x}}{\left (\left (75694080 x^{4}-340623360 x^{3}+549077760 x^{2}-377952960 x +93302055\right ) \operatorname {KummerM}\left (\frac {1}{4}, 4, 4 x -4\right )+70963200 \left (x^{3}-\frac {21}{8} x^{2}+\frac {293}{128} x -\frac {663}{1024}\right ) \operatorname {KummerM}\left (-\frac {3}{4}, 4, 4 x -4\right )\right ) \operatorname {KummerU}\left (\frac {21}{4}, 4, 4 x -4\right )+67108864 \operatorname {KummerM}\left (\frac {21}{4}, 4, 4 x -4\right ) \left (\left (x^{4}-\frac {9}{2} x^{3}+\frac {1857}{256} x^{2}-\frac {5113}{1024} x +\frac {80781}{65536}\right ) \operatorname {KummerU}\left (\frac {1}{4}, 4, 4 x -4\right )-\frac {\left (x^{3}-\frac {21}{8} x^{2}+\frac {293}{128} x -\frac {663}{1024}\right ) \operatorname {KummerU}\left (-\frac {3}{4}, 4, 4 x -4\right )}{4}\right )}d x \right )-540540 \operatorname {KummerU}\left (\frac {21}{4}, 4, 4 x -4\right ) \left (\int \frac {\cos \left (x \right ) \operatorname {KummerM}\left (\frac {21}{4}, 4, 4 x -4\right ) {\mathrm e}^{4 x}}{16384 \left (\left (-1024 x^{3}+2688 x^{2}-2344 x +663\right ) \operatorname {KummerU}\left (-\frac {3}{4}, 4, 4 x -4\right )+4096 \left (x^{4}-\frac {9}{2} x^{3}+\frac {1857}{256} x^{2}-\frac {5113}{1024} x +\frac {80781}{65536}\right ) \operatorname {KummerU}\left (\frac {1}{4}, 4, 4 x -4\right )\right ) \operatorname {KummerM}\left (\frac {21}{4}, 4, 4 x -4\right )+75694080 \operatorname {KummerU}\left (\frac {21}{4}, 4, 4 x -4\right ) \left (\left (x^{4}-\frac {9}{2} x^{3}+\frac {1857}{256} x^{2}-\frac {5113}{1024} x +\frac {80781}{65536}\right ) \operatorname {KummerM}\left (\frac {1}{4}, 4, 4 x -4\right )+\frac {15 \left (x^{3}-\frac {21}{8} x^{2}+\frac {293}{128} x -\frac {663}{1024}\right ) \operatorname {KummerM}\left (-\frac {3}{4}, 4, 4 x -4\right )}{16}\right )}d x \right )+\operatorname {KummerU}\left (\frac {21}{4}, 4, 4 x -4\right ) c_1 +\operatorname {KummerM}\left (\frac {21}{4}, 4, 4 x -4\right ) c_2 \right )
\]
✓ Solution by Mathematica
Time used: 7.201 (sec). Leaf size: 230
DSolve[(1-x)*D[y[x],{x,2}]-4*x*D[y[x],x]+5*y[x]==Cos[x],y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to e^{-4 x} \left (\operatorname {HypergeometricU}\left (\frac {21}{4},4,4 x-4\right ) \int _1^x\frac {e^{4 K[1]} \cos (K[1]) L_{-\frac {21}{4}}^3(4 K[1]-4)}{(K[1]-1) \left (21 \operatorname {HypergeometricU}\left (\frac {25}{4},5,4 K[1]-4\right ) L_{-\frac {21}{4}}^3(4 K[1]-4)-4 \operatorname {HypergeometricU}\left (\frac {21}{4},4,4 K[1]-4\right ) L_{-\frac {25}{4}}^4(4 K[1]-4)\right )}dK[1]+L_{-\frac {21}{4}}^3(4 x-4) \int _1^x\frac {e^{4 K[2]} \cos (K[2]) \operatorname {HypergeometricU}\left (\frac {21}{4},4,4 K[2]-4\right )}{(K[2]-1) \left (4 \operatorname {HypergeometricU}\left (\frac {21}{4},4,4 K[2]-4\right ) L_{-\frac {25}{4}}^4(4 K[2]-4)-21 \operatorname {HypergeometricU}\left (\frac {25}{4},5,4 K[2]-4\right ) L_{-\frac {21}{4}}^3(4 K[2]-4)\right )}dK[2]+c_1 \operatorname {HypergeometricU}\left (\frac {21}{4},4,4 x-4\right )+c_2 L_{-\frac {21}{4}}^3(4 x-4)\right )
\]