Internal problem ID [4214]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime }+y-x^{\frac {3}{2}}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✗ 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: 0.392 (sec). Leaf size: 688
AsymptoticDSolveValue[x^3*y''[x]+y[x]==x^(3/2),y[x],{x,0,5}]
\[ y(x)\to \frac {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 ) \left (e^{-\frac {2 i}{\sqrt {x}}} \sqrt [4]{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 )-(2547645096841445376-2547645096841445376 i) \sqrt {\pi } \text {Erf}\left (\frac {1+i}{\sqrt [4]{x}}\right )\right )}{1801439850948198400}+\frac {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 ) \left (e^{\frac {2 i}{\sqrt {x}}} \sqrt [4]{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 )-(2547645096841445376-2547645096841445376 i) \sqrt {\pi } \text {Erfi}\left (\frac {1+i}{\sqrt [4]{x}}\right )\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 ) \]