4.44.44 \(x^3 y'''(x)+x^2 \log (x) y''(x)+2 x y'(x)-y(x)=2 x^3\)

ODE
\[ x^3 y'''(x)+x^2 \log (x) y''(x)+2 x y'(x)-y(x)=2 x^3 \] ODE Classification

[[_3rd_order, _linear, _nonhomogeneous]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.361892 (sec), leaf count = 105

\[\left \{\left \{y(x)\to \frac {1}{6} x^2 e^{\frac {1}{2} \left (-\log ^2(x)-5\right )} \left (3 \sqrt {2 e \pi } c_2 \text {erfi}\left (\frac {\log (x)-2}{\sqrt {2}}\right )+e^2 \left (\sqrt {2 \pi } \text {erfi}\left (\frac {\log (x)+1}{\sqrt {2}}\right )+3 \sqrt {2 \pi } c_3 \text {erfi}\left (\frac {\log (x)-1}{\sqrt {2}}\right )+6 \sqrt {e} c_1\right )\right )\right \}\right \}\]

Maple ✓
cpu = 1.231 (sec), leaf count = 211

\[\left [y \left (x \right ) = \left (\int i \sqrt {\pi }\, \left (\frac {\erf \left (\frac {i \sqrt {2}\, \left (\ln \left (x \right )-1\right )}{2}\right ) {\mathrm e}^{\frac {\ln \left (x \right )^{2}}{2}} {\mathrm e}^{2}}{x^{2}}-\frac {\erf \left (\frac {i \sqrt {2}\, \left (-2+\ln \left (x \right )\right )}{2}\right ) {\mathrm e}^{\frac {\ln \left (x \right )^{2}}{2}} {\mathrm e}^{\frac {1}{2}}}{x}\right ) \sqrt {2}\, x^{3} {\mathrm e}^{-\frac {\ln \left (x \right )^{2}}{2}} {\mathrm e}^{-\frac {5}{2}}d x \right ) {\mathrm e}^{-\frac {\ln \left (x \right )^{2}}{2}} x^{2}+\frac {i x^{5} \sqrt {\pi }\, \sqrt {2}\, {\mathrm e}^{-2} \erf \left (\frac {i \sqrt {2}\, \left (-2+\ln \left (x \right )\right )}{2}\right ) {\mathrm e}^{-\frac {\ln \left (x \right )^{2}}{2}}}{3}-\frac {i x^{4} \sqrt {\pi }\, \sqrt {2}\, {\mathrm e}^{-\frac {1}{2}} \erf \left (\frac {i \sqrt {2}\, \left (\ln \left (x \right )-1\right )}{2}\right ) {\mathrm e}^{-\frac {\ln \left (x \right )^{2}}{2}}}{2}+\textit {\_C1} \,{\mathrm e}^{-\frac {\ln \left (x \right )^{2}}{2}} x^{2}+\textit {\_C2} \erf \left (\frac {i \sqrt {2}\, \left (-2+\ln \left (x \right )\right )}{2}\right ) {\mathrm e}^{-\frac {\ln \left (x \right )^{2}}{2}} x^{2}+\textit {\_C3} \erf \left (\frac {i \sqrt {2}\, \left (\ln \left (x \right )-1\right )}{2}\right ) {\mathrm e}^{-\frac {\ln \left (x \right )^{2}}{2}} x^{2}\right ]\] Mathematica raw input

DSolve[-y[x] + 2*x*y'[x] + x^2*Log[x]*y''[x] + x^3*y'''[x] == 2*x^3,y[x],x]

Mathematica raw output

{{y[x] -> (E^((-5 - Log[x]^2)/2)*x^2*(3*Sqrt[2*E*Pi]*C[2]*Erfi[(-2 + Log[x])/Sqr
t[2]] + E^2*(6*Sqrt[E]*C[1] + 3*Sqrt[2*Pi]*C[3]*Erfi[(-1 + Log[x])/Sqrt[2]] + Sq
rt[2*Pi]*Erfi[(1 + Log[x])/Sqrt[2]])))/6}}

Maple raw input

dsolve(x^3*diff(diff(diff(y(x),x),x),x)+x^2*diff(diff(y(x),x),x)*ln(x)+2*x*diff(y(x),x)-y(x) = 2*x^3, y(x))

Maple raw output

[y(x) = Int(I*Pi^(1/2)*(erf(1/2*I*2^(1/2)*(ln(x)-1))*exp(1/2*ln(x)^2)/x^2*exp(2)
-erf(1/2*I*2^(1/2)*(-2+ln(x)))*exp(1/2*ln(x)^2)/x*exp(1/2))*2^(1/2)*x^3*exp(-1/2
*ln(x)^2)*exp(-5/2),x)*exp(-1/2*ln(x)^2)*x^2+1/3*I*x^5*Pi^(1/2)*2^(1/2)*exp(-2)*
erf(1/2*I*2^(1/2)*(-2+ln(x)))*exp(-1/2*ln(x)^2)-1/2*I*x^4*Pi^(1/2)*2^(1/2)*exp(-
1/2)*erf(1/2*I*2^(1/2)*(ln(x)-1))*exp(-1/2*ln(x)^2)+_C1*exp(-1/2*ln(x)^2)*x^2+_C
2*erf(1/2*I*2^(1/2)*(-2+ln(x)))*exp(-1/2*ln(x)^2)*x^2+_C3*erf(1/2*I*2^(1/2)*(ln(
x)-1))*exp(-1/2*ln(x)^2)*x^2]