34.4.12 problem 12
Internal
problem
ID
[6066]
Book
:
A
treatise
on
ordinary
and
partial
differential
equations
by
William
Woolsey
Johnson.
1913
Section
:
Chapter
VII,
Solutions
in
series.
Examples
XV.
page
194
Problem
number
:
12
Date
solved
:
Monday, January 27, 2025 at 01:34:46 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} x^{3} y^{\prime \prime }+y&=x^{{3}/{2}} \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✗ Solution by Maple
Order:=6;
dsolve(x^3*diff(y(x),x$2)+y(x)=x^(3/2),y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.366 (sec). Leaf size: 740
AsymptoticDSolveValue[x^3*D[y[x],{x,2}]+y[x]==x^(3/2),y[x],{x,0,"6"-1}]
\[
y(x)\to \frac {x \left (\frac {468131288625 i x^{9/2}}{8796093022208}-\frac {66891825 i x^{7/2}}{4294967296}+\frac {72765 i x^{5/2}}{8388608}-\frac {105 i x^{3/2}}{8192}+\frac {33424574007825 x^5}{281474976710656}-\frac {14783093325 x^4}{549755813888}+\frac {2837835 x^3}{268435456}-\frac {4725 x^2}{524288}+\frac {15 x}{512}+\frac {3 i \sqrt {x}}{16}+1\right ) \left (\frac {2547645096841445376 (-1)^{3/4} \sqrt {2 \pi } e^{\frac {2 i}{\sqrt {x}}} \text {erf}\left (\frac {\sqrt [4]{-1} \sqrt {2}}{\sqrt [4]{x}}\right )}{\sqrt [4]{x}}-2540267624594700 x^{11/2}+14482858554964800 x^{9/2}-6169315551759360 x^{7/2}+4596814259896320 x^{5/2}-9826098960138240 x^{3/2}-14606538841419525 i x^6+20856934180882800 i x^5-9106700860857600 i x^4+4828156832378880 i x^3-5650801088593920 i x^2+28971502421409792 i x+263808651263737856 \sqrt {x}-\frac {2547645096841445376 (-1)^{3/4} \sqrt {2 \pi } e^{\frac {2 i}{\sqrt {x}}}}{\sqrt [4]{x}}+2547645096841445376 i\right )}{1801439850948198400}+\frac {e^{-\frac {2 i}{\sqrt {x}}} x \left (-\frac {468131288625 i x^{9/2}}{8796093022208}+\frac {66891825 i x^{7/2}}{4294967296}-\frac {72765 i x^{5/2}}{8388608}+\frac {105 i x^{3/2}}{8192}+\frac {33424574007825 x^5}{281474976710656}-\frac {14783093325 x^4}{549755813888}+\frac {2837835 x^3}{268435456}-\frac {4725 x^2}{524288}+\frac {15 x}{512}-\frac {3 i \sqrt {x}}{16}+1\right ) \left (\frac {2547645096841445376 \sqrt [4]{-1} \sqrt {2 \pi } \text {erf}\left (\frac {(-1)^{3/4} \sqrt {2}}{\sqrt [4]{x}}\right )}{\sqrt [4]{x}}+e^{\frac {2 i}{\sqrt {x}}} \left (-2540267624594700 x^{11/2}+14482858554964800 x^{9/2}-6169315551759360 x^{7/2}+4596814259896320 x^{5/2}-9826098960138240 x^{3/2}+14606538841419525 i x^6-20856934180882800 i x^5+9106700860857600 i x^4-4828156832378880 i x^3+5650801088593920 i x^2-28971502421409792 i x+263808651263737856 \sqrt {x}-2547645096841445376 i\right )+\frac {2547645096841445376 \sqrt [4]{-1} \sqrt {2 \pi }}{\sqrt [4]{x}}\right )}{1801439850948198400}+c_1 e^{-\frac {2 i}{\sqrt {x}}} x^{3/4} \left (-\frac {468131288625 i x^{9/2}}{8796093022208}+\frac {66891825 i x^{7/2}}{4294967296}-\frac {72765 i x^{5/2}}{8388608}+\frac {105 i x^{3/2}}{8192}+\frac {33424574007825 x^5}{281474976710656}-\frac {14783093325 x^4}{549755813888}+\frac {2837835 x^3}{268435456}-\frac {4725 x^2}{524288}+\frac {15 x}{512}-\frac {3 i \sqrt {x}}{16}+1\right )+c_2 e^{\frac {2 i}{\sqrt {x}}} x^{3/4} \left (\frac {468131288625 i x^{9/2}}{8796093022208}-\frac {66891825 i x^{7/2}}{4294967296}+\frac {72765 i x^{5/2}}{8388608}-\frac {105 i x^{3/2}}{8192}+\frac {33424574007825 x^5}{281474976710656}-\frac {14783093325 x^4}{549755813888}+\frac {2837835 x^3}{268435456}-\frac {4725 x^2}{524288}+\frac {15 x}{512}+\frac {3 i \sqrt {x}}{16}+1\right )
\]