Internal problem ID [11321]
Internal file name [OUTPUT/10307_Tuesday_December_27_2022_04_06_10_AM_81835214/index.tex]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IX, Miscellaneous methods for solving equations of higher order than first.
Article 59. Linear equations with particular integral known. Page 136
Problem number: Ex 2.
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, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {x y^{\prime \prime \prime }-y^{\prime \prime }-x y^{\prime }+y=-x^{2}+1} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime \prime }\right ) x -\frac {d}{d x}y^{\prime }-x y^{\prime }+y=-x^{2}+1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \end {array} \]
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 3; linear nonhomogeneous with symmetry [0,1] trying high order linear exact nonhomogeneous trying differential order: 3; missing the dependent variable checking if the LODE is of Euler type Equation is the LCLM of -1/x*y(x)+diff(y(x),x), y(x)+diff(y(x),x), -y(x)+diff(y(x),x) trying differential order: 1; missing the dependent variable checking if the LODE is of Euler type <- LODE of Euler type successful Euler equation successful trying differential order: 1; missing the dependent variable checking if the LODE has constant coefficients <- constant coefficients successful trying differential order: 1; missing the dependent variable checking if the LODE has constant coefficients <- constant coefficients successful <- solving the LCLM ode successful `
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 22
dsolve(x*diff(y(x),x$3)-diff(y(x),x$2)-x*diff(y(x),x)+y(x)=1-x^2,y(x), singsol=all)
\[ y \left (x \right ) = x^{2}+3+c_{1} x +c_{2} {\mathrm e}^{x}+c_{3} {\mathrm e}^{-x} \]
✓ Solution by Mathematica
Time used: 0.242 (sec). Leaf size: 28
DSolve[x*y'''[x]-y''[x]-x*y'[x]+y[x]==1-x^2,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x^2+c_1 x-c_2 \cosh (x)+i c_3 \sinh (x)+3 \]