60.4.33 problem 1481

Internal problem ID [11484]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1481
Date solved : Monday, January 27, 2025 at 11:22:34 PM
CAS classification : [[_3rd_order, _fully, _exact, _linear]]

\begin{align*} x y^{\prime \prime \prime }+\left (x^{2}-3\right ) y^{\prime \prime }+4 x y^{\prime }+2 y-f \left (x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 40

dsolve(x*diff(diff(diff(y(x),x),x),x)+(x^2-3)*diff(diff(y(x),x),x)+4*x*diff(y(x),x)+2*y(x)-f(x)=0,y(x), singsol=all)
 

\[ y = \left (c_3 +\int \frac {\left (2 c_{1} x +c_{2} +\int \left (\int fd x \right )d x \right ) {\mathrm e}^{\frac {x^{2}}{2}}}{x^{6}}d x \right ) {\mathrm e}^{-\frac {x^{2}}{2}} x^{5} \]

✓ Solution by Mathematica

Time used: 0.354 (sec). Leaf size: 346

DSolve[-f[x] + 2*y[x] + 4*x*D[y[x],x] + (-3 + x^2)*D[y[x],{x,2}] + x*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{240} \left (240 e^{-\frac {x^2}{2}} x^5 \int _1^x\frac {f(K[1]) \left (8 \sqrt {2 \pi } \text {erfi}\left (\frac {K[1]}{\sqrt {2}}\right ) K[1]^5-15 \operatorname {ExpIntegralEi}\left (\frac {K[1]^2}{2}\right ) K[1]^4+2 e^{\frac {K[1]^2}{2}} \left (-8 K[1]^4+7 K[1]^2+6\right )\right )}{240 K[1]^4}dK[1]+8 \sqrt {2 \pi } e^{-\frac {x^2}{2}} x^5 \text {erfi}\left (\frac {x}{\sqrt {2}}\right ) \int _1^x-f(K[2]) K[2]dK[2]+15 x \left (e^{-\frac {x^2}{2}} x^4 \operatorname {ExpIntegralEi}\left (\frac {x^2}{2}\right )-2 \left (x^2+2\right )\right ) \int _1^xf(K[3])dK[3]-16 x^4 \int _1^x-f(K[2]) K[2]dK[2]-16 x^2 \int _1^x-f(K[2]) K[2]dK[2]-48 \int _1^x-f(K[2]) K[2]dK[2]+8 \sqrt {2 \pi } c_2 e^{-\frac {x^2}{2}} x^5 \text {erfi}\left (\frac {x}{\sqrt {2}}\right )+15 c_3 e^{-\frac {x^2}{2}} x^5 \operatorname {ExpIntegralEi}\left (\frac {x^2}{2}\right )-16 c_2 x^4-30 c_3 x^3-16 c_2 x^2+240 c_1 e^{-\frac {x^2}{2}} x^5-60 c_3 x-48 c_2\right ) \]