1.47 problem 48

1.47.1 Maple step by step solution

Internal problem ID [3192]
Internal file name [OUTPUT/2684_Sunday_June_05_2022_08_38_45_AM_41111291/index.tex]

Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number: 48.
ODE order: 1.
ODE degree: 1.

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

Maple gives the following as the ode type

[_rational]

Unable to solve or complete the solution.

\[ \boxed {2 x y^{4}-y+\left (4 x^{3} y^{3}-x \right ) y^{\prime }=0} \] Unable to determine ODE type.

1.47.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x y^{4}-y+\left (4 x^{3} y^{3}-x \right ) y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-2 x y^{4}+y}{4 x^{3} y^{3}-x} \end {array} \]

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 
equivalence obtained to this Abel ODE: diff(y(x),x) = -3*y(x)/x+(-16*x+6)*y(x)^2+(-48*x^3+24*x^2)*y(x)^3 
trying to solve the Abel ODE ... 
Looking for potential symmetries 
Looking for potential symmetries 
differential order: 1; trying a linearization to 2nd order 
--- trying a change of variables {x -> y(x), y(x) -> x} 
differential order: 1; trying a linearization to 2nd order 
trying 1st order ODE linearizable_by_differentiation 
--- Trying Lie symmetry methods, 1st order --- 
`, `-> Computing symmetries using: way = 2 
`, `-> Computing symmetries using: way = 3 
`, `-> Computing symmetries using: way = 4 
trying symmetry patterns for 1st order ODEs 
-> trying a symmetry pattern of the form [F(x)*G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)*G(y)] 
-> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] 
-> trying a symmetry pattern of the form [F(x),G(x)] 
-> trying a symmetry pattern of the form [F(y),G(y)] 
-> trying a symmetry pattern of the form [F(x)+G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)+G(y)] 
-> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
-> trying a symmetry pattern of conformal type`
 

Solution by Maple

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

\[ \text {No solution found} \]

Solution by Mathematica

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

DSolve[(2*x*y[x]^4-y[x])+(4*x^3*y[x]^3-x)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

Not solved