2.1.91 problem 89

Internal problem ID [8479]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 89
Date solved : Wednesday, December 18, 2024 at 01:53:08 AM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

Solve

\begin{align*} y^{\prime \prime }-y^{\prime } y&=2 x \end{align*}

Maple step by step solution

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
trying 2nd order WeierstrassP 
trying 2nd order JacobiSN 
differential order: 2; trying a linearization to 3rd order 
trying 2nd order ODE linearizable_by_differentiation 
trying 2nd order, 2 integrating factors of the form mu(x,y) 
trying differential order: 2; missing variables 
-> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 
   --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- 
   -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y) 
trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases 
trying symmetries linear in x and y(x) 
trying differential order: 2; exact nonlinear 
-> Calling odsolve with the ODE`, diff(_b(_a), _a) = (1/2)*_b(_a)^2+_a^2-c__1, _b(_a)`   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying a quadrature 
   trying 1st order linear 
   trying Bernoulli 
   trying separable 
   trying inverse linear 
   trying homogeneous types: 
   trying Chini 
   differential order: 1; looking for linear symmetries 
   trying exact 
   Looking for potential symmetries 
   trying Riccati 
   trying Riccati Special 
   trying Riccati sub-methods: 
      trying Riccati to 2nd Order 
      -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = (-(1/2)*x^2+(1/2)*c__1)*y(x), y(x)`         *** Sublevel 3 *** 
         Methods for second order ODEs: 
         --- Trying classification methods --- 
         trying a quadrature 
         checking if the LODE has constant coefficients 
         checking if the LODE is of Euler type 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         checking if the LODE is missing y 
         -> Trying a Liouvillian solution using Kovacics algorithm 
         <- No Liouvillian solutions exists 
         -> Trying a solution in terms of special functions: 
            -> Bessel 
            -> elliptic 
            -> Legendre 
            -> Whittaker 
               -> hyper3: Equivalence to 1F1 under a power @ Moebius 
               <- hyper3 successful: received ODE is equivalent to the 1F1 ODE 
            <- Whittaker successful 
         <- special function solution successful 
      <- Riccati to 2nd Order successful 
<- differential order: 2; exact nonlinear successful`
 

Maple dsolve solution

Solving time : 0.093 (sec)
Leaf size : 147

dsolve(diff(diff(y(x),x),x)-y(x)*diff(y(x),x) = 2*x, 
       y(x),singsol=all)
 

\[ y = \frac {-\operatorname {WhittakerM}\left (\frac {i c_{1} \sqrt {2}}{8}+1, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right ) \left (6+i c_{1} \sqrt {2}\right )+8 c_{2} \operatorname {WhittakerW}\left (\frac {i c_{1} \sqrt {2}}{8}+1, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+2 \left (1-i \left (x^{2}-\frac {c_{1}}{2}\right ) \sqrt {2}\right ) \left (c_{2} \operatorname {WhittakerW}\left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+\operatorname {WhittakerM}\left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )\right )}{2 x \left (c_{2} \operatorname {WhittakerW}\left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )+\operatorname {WhittakerM}\left (\frac {i c_{1} \sqrt {2}}{8}, \frac {1}{4}, \frac {i \sqrt {2}\, x^{2}}{2}\right )\right )} \]

Mathematica DSolve solution

Solving time : 45.186 (sec)
Leaf size : 318

DSolve[{D[y[x],{x,2}]+D[y[x],x]*y[x]==2*x,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\sqrt [4]{2} \left (\sqrt [4]{2} x \operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (-\sqrt {2} c_1-2\right ),i \sqrt [4]{2} x\right )+2 i \operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (2-\sqrt {2} c_1\right ),i \sqrt [4]{2} x\right )+c_2 \left (2 \operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (\sqrt {2} c_1+2\right ),\sqrt [4]{2} x\right )-\sqrt [4]{2} x \operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (\sqrt {2} c_1-2\right ),\sqrt [4]{2} x\right )\right )\right )}{\operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (-\sqrt {2} c_1-2\right ),i \sqrt [4]{2} x\right )+c_2 \operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (\sqrt {2} c_1-2\right ),\sqrt [4]{2} x\right )} \\ y(x)\to \sqrt {2} x-\frac {2 \sqrt [4]{2} \operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (\sqrt {2} c_1+2\right ),\sqrt [4]{2} x\right )}{\operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (\sqrt {2} c_1-2\right ),\sqrt [4]{2} x\right )} \\ y(x)\to \sqrt {2} x-\frac {2 \sqrt [4]{2} \operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (\sqrt {2} c_1+2\right ),\sqrt [4]{2} x\right )}{\operatorname {ParabolicCylinderD}\left (\frac {1}{4} \left (\sqrt {2} c_1-2\right ),\sqrt [4]{2} x\right )} \\ \end{align*}