Internal problem ID [6139]
Internal file name [OUTPUT/5387_Sunday_June_05_2022_03_35_57_PM_84684605/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.3 SEPARABLE EQUATIONS. Page
12
Problem number: 1(a).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {x^{5} y^{\prime }+y^{5}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {y^{5}}{x^{5}} \end {align*}
Where \(f(x)=-\frac {1}{x^{5}}\) and \(g(y)=y^{5}\). Integrating both sides gives \begin{align*} \frac {1}{y^{5}} \,dy &= -\frac {1}{x^{5}} \,d x \\ \int { \frac {1}{y^{5}} \,dy} &= \int {-\frac {1}{x^{5}} \,d x} \\ -\frac {1}{4 y^{4}}&=\frac {1}{4 x^{4}}+c_{1} \\ \end{align*} The solution is \[ -\frac {1}{4 y^{4}}-\frac {1}{4 x^{4}}-c_{1} = 0 \]
The solution(s) found are the following \begin{align*} \tag{1} -\frac {1}{4 y^{4}}-\frac {1}{4 x^{4}}-c_{1} &= 0 \\ \end{align*}
Verification of solutions
\[ -\frac {1}{4 y^{4}}-\frac {1}{4 x^{4}}-c_{1} = 0 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{5} y^{\prime }+y^{5}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {y^{5}}{x^{5}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{5}}=-\frac {1}{x^{5}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{5}}d x =\int -\frac {1}{x^{5}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{4 y^{4}}=\frac {1}{4 x^{4}}+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 67
dsolve(x^5*diff(y(x),x)+y(x)^5=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {x}{\left (c_{1} x^{4}-1\right )^{\frac {1}{4}}} \\ y \left (x \right ) &= -\frac {x}{\left (c_{1} x^{4}-1\right )^{\frac {1}{4}}} \\ y \left (x \right ) &= \frac {x}{\sqrt {-\sqrt {c_{1} x^{4}-1}}} \\ y \left (x \right ) &= -\frac {x}{\sqrt {-\sqrt {c_{1} x^{4}-1}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.489 (sec). Leaf size: 145
DSolve[x^5*y'[x]+y[x]^5==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x}{\sqrt [4]{-1-4 c_1 x^4}} \\ y(x)\to -\frac {i x}{\sqrt [4]{-1-4 c_1 x^4}} \\ y(x)\to \frac {i x}{\sqrt [4]{-1-4 c_1 x^4}} \\ y(x)\to \frac {x}{\sqrt [4]{-1-4 c_1 x^4}} \\ y(x)\to 0 \\ y(x)\to -\frac {(1+i) x}{\sqrt {2}} \\ y(x)\to -\frac {(1-i) x}{\sqrt {2}} \\ y(x)\to \frac {(1-i) x}{\sqrt {2}} \\ y(x)\to \frac {(1+i) x}{\sqrt {2}} \\ \end{align*}