Internal problem ID [9807]
Internal file name [OUTPUT/8750_Monday_June_06_2022_05_23_32_AM_28665111/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1481.
ODE order: 3.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _fully, _exact, _linear]]
Unable to solve or complete the solution.
\[ \boxed {x y^{\prime \prime \prime }+\left (x^{2}-3\right ) y^{\prime \prime }+4 y^{\prime } x +2 y=f \left (x \right )} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (\frac {d}{d x}y^{\prime \prime }\right )+\left (x^{2}-3\right ) \left (\frac {d}{d x}y^{\prime }\right )+4 y^{\prime } x +2 y=f \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \end {array} \]
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable <- high order exact linear fully integrable successful`
✓ Solution by Maple
Time used: 0.0 (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)
\[ y \left (x \right ) = \left (c_{3} +\int \frac {\left (2 c_{1} x +c_{2} +\int \int f \left (x \right )d x d x \right ) {\mathrm e}^{\frac {x^{2}}{2}}}{x^{6}}d x \right ) {\mathrm e}^{-\frac {x^{2}}{2}} x^{5} \]
✓ Solution by Mathematica
Time used: 0.31 (sec). Leaf size: 346
DSolve[-f[x] + 2*y[x] + 4*x*y'[x] + (-3 + x^2)*y''[x] + x*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{240} \left (240 e^{-\frac {x^2}{2}} x^5 \int _1^x\frac {f(K[1]) \left (8 \sqrt {2 \pi } \text {erfi}\left (\frac {K[1]}{\sqrt {2}}\right ) K[1]^5-15 \operatorname {ExpIntegralEi}\left (\frac {K[1]^2}{2}\right ) K[1]^4+2 e^{\frac {K[1]^2}{2}} \left (-8 K[1]^4+7 K[1]^2+6\right )\right )}{240 K[1]^4}dK[1]+8 \sqrt {2 \pi } e^{-\frac {x^2}{2}} x^5 \text {erfi}\left (\frac {x}{\sqrt {2}}\right ) \int _1^x-f(K[2]) K[2]dK[2]+15 x \left (e^{-\frac {x^2}{2}} x^4 \operatorname {ExpIntegralEi}\left (\frac {x^2}{2}\right )-2 \left (x^2+2\right )\right ) \int _1^xf(K[3])dK[3]-16 x^4 \int _1^x-f(K[2]) K[2]dK[2]-16 x^2 \int _1^x-f(K[2]) K[2]dK[2]-48 \int _1^x-f(K[2]) K[2]dK[2]+8 \sqrt {2 \pi } c_2 e^{-\frac {x^2}{2}} x^5 \text {erfi}\left (\frac {x}{\sqrt {2}}\right )+15 c_3 e^{-\frac {x^2}{2}} x^5 \operatorname {ExpIntegralEi}\left (\frac {x^2}{2}\right )-16 c_2 x^4-30 c_3 x^3-16 c_2 x^2+240 c_1 e^{-\frac {x^2}{2}} x^5-60 c_3 x-48 c_2\right ) \]