1.28 problem 28

Internal problem ID [1897]
Internal file name [OUTPUT/1898_Sunday_June_05_2022_02_38_09_AM_33704393/index.tex]

Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section: Exercise 5, page 21
Problem number: 28.
ODE order: 1.
ODE degree: 1.

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

Maple gives the following as the ode type

[_rational, _Abel]

Unable to solve or complete the solution.

\[ \boxed {3 y^{\prime } x -y^{3}-2 y=-x^{2}} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}

1.28.1 Existence and uniqueness analysis

This is non linear first order ODE. In canonical form it is written as \begin {align*} y^{\prime } &= f(x,y)\\ &= \frac {y^{3}-x^{2}+2 y}{3 x} \end {align*}

The \(x\) domain of \(f(x,y)\) when \(y=1\) is \[ \{x <0\boldsymbol {\lor }0

The \(x\) domain of \(\frac {\partial f}{\partial y}\) when \(y=1\) is \[ \{x <0\boldsymbol {\lor }0

1.28.2 Solving as abelFirstKind ode

This is Abel first kind ODE, it has the form \[ y^{\prime }= f_0(x)+f_1(x) y +f_2(x)y^{2}+f_3(x)y^{3} \] Comparing the above to given ODE which is \begin {align*} y^{\prime }&=\frac {y^{3}}{3 x}+\frac {2 y}{3 x}-\frac {x}{3}\tag {1} \end {align*}

Therefore \begin {align*} f_0(x) &= -\frac {x}{3}\\ f_1(x) &= \frac {2}{3 x}\\ f_2(x) &= 0\\ f_3(x) &= \frac {1}{3 x} \end {align*}

Since \(f_2(x)=0\) then we check the Abel invariant to see if it depends on \(x\) or not. The Abel invariant is given by \begin {align*} -\frac {f_{1}^{3}}{f_{0}^{2} f_{3}} \end {align*}

Which when evaluating gives \begin {align*} 0 \end {align*}

Since the Abel invariant does not depend on \(x\) then this ode can be solved directly.

1.28.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [3 y^{\prime } x -y^{3}-2 y=-x^{2}, y \left (1\right )=1\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-x^{2}+y^{3}+2 y}{3 x} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (1\right )=1 \\ {} & {} & 0 \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} 0 \\ {} & {} & 0=0 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} 0=0\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & 0 \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & 0 \end {array} \]

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 
<- Chini successful`
 

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 7

dsolve([x^2+3*x*diff(y(x),x)=y(x)^3+2*y(x),y(1) = 1],y(x), singsol=all)
 

\[ y \left (x \right ) = x^{\frac {2}{3}} \]

✗ Solution by Mathematica

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

DSolve[{x^2+3*x*y'[x]==y[x]^3+2*y[x],y[1]==1},y[x],x,IncludeSingularSolutions -> True]
 

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