1.2 problem 2

Internal problem ID [14045]
Internal file name [OUTPUT/13726_Friday_March_01_2024_09_08_48_AM_93797961/index.tex]

Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section: Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number: 2.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "unknown"

Maple gives the following as the ode type

[`y=_G(x,y')`]

Unable to solve or complete the solution.

\[ \boxed {y y^{\prime }+y^{4}=\sin \left (x \right )} \] Unable to determine ODE type.

1.2.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }+y^{4}=\sin \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-y^{4}+\sin \left (x \right )}{y} \end {array} \]

Maple trace

`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 inverse_Riccati 
trying an equivalence to an Abel ODE 
differential order: 1; trying a linearization to 2nd order 
-> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = 4*sin(x)*y(x), y(x)`   *** Sublevel 2 *** 
   Methods for second order ODEs: 
   --- Trying classification methods --- 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   checking if the LODE is missing y 
   -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
   -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
   -> Trying changes of variables to rationalize or make the ODE simpler 
      trying a symmetry of the form [xi=0, eta=F(x)] 
      checking if the LODE is missing y 
      -> Trying an equivalence, under non-integer power transformations, 
         to LODEs admitting Liouvillian solutions. 
         -> 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 
         -> hypergeometric 
            -> heuristic approach 
            -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
         -> Mathieu 
            -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius 
            Equivalence transformation and function parameters: {t = 1/2*t+1/2}, {kappa = 60, mu = 128} 
            <- Equivalence to the rational form of Mathieu ODE successful 
         <- Mathieu successful 
      <- special function solution successful 
      Change of variables used: 
         [x = arccos(t)] 
      Linear ODE actually solved: 
         -4*(-t^2+1)^(1/2)*u(t)-t*diff(u(t),t)+(-t^2+1)*diff(diff(u(t),t),t) = 0 
   <- change of variables successful 
<- differential order: 1; linearization to 2nd order successful`
 

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 164

dsolve(y(x)*diff(y(x),x)+y(x)^4=sin(x),y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -\frac {\sqrt {\left (c_{1} \operatorname {MathieuC}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )+\operatorname {MathieuS}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )\right ) \left (c_{1} \operatorname {MathieuCPrime}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )+\operatorname {MathieuSPrime}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )\right )}}{2 c_{1} \operatorname {MathieuC}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )+2 \operatorname {MathieuS}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )} \\ y \left (x \right ) &= \frac {\sqrt {\left (c_{1} \operatorname {MathieuC}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )+\operatorname {MathieuS}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )\right ) \left (c_{1} \operatorname {MathieuCPrime}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )+\operatorname {MathieuSPrime}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )\right )}}{2 c_{1} \operatorname {MathieuC}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )+2 \operatorname {MathieuS}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )} \\ \end{align*}

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

DSolve[y[x]*y'[x]+y[x]^4==Sin[x],y[x],x,IncludeSingularSolutions -> True]
 

Not solved