Internal problem ID [15103]
Internal file name [OUTPUT/15104_Sunday_April_21_2024_01_29_51_PM_25456079/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 8. First order not solved for the derivative. Exercises page 67
Problem number: 216.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y^{\frac {2}{5}}+{y^{\prime }}^{\frac {2}{5}}=a^{\frac {2}{5}}} \]
Integrating both sides gives \begin{align*} \int \frac {1}{\left (-y^{\frac {2}{5}}+a^{\frac {2}{5}}\right )^{\frac {5}{2}}}d y &= \int d x \\ \frac {5 y^{\frac {3}{5}}}{3 \left (-y^{\frac {2}{5}}+a^{\frac {2}{5}}\right )^{\frac {3}{2}}}-\frac {5 y^{\frac {1}{5}}}{\sqrt {-y^{\frac {2}{5}}+a^{\frac {2}{5}}}}+5 \arctan \left (\frac {y^{\frac {1}{5}}}{\sqrt {-y^{\frac {2}{5}}+a^{\frac {2}{5}}}}\right )&=x +c_{1} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} \frac {5 y^{\frac {3}{5}}}{3 \left (-y^{\frac {2}{5}}+a^{\frac {2}{5}}\right )^{\frac {3}{2}}}-\frac {5 y^{\frac {1}{5}}}{\sqrt {-y^{\frac {2}{5}}+a^{\frac {2}{5}}}}+5 \arctan \left (\frac {y^{\frac {1}{5}}}{\sqrt {-y^{\frac {2}{5}}+a^{\frac {2}{5}}}}\right ) &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \frac {5 y^{\frac {3}{5}}}{3 \left (-y^{\frac {2}{5}}+a^{\frac {2}{5}}\right )^{\frac {3}{2}}}-\frac {5 y^{\frac {1}{5}}}{\sqrt {-y^{\frac {2}{5}}+a^{\frac {2}{5}}}}+5 \arctan \left (\frac {y^{\frac {1}{5}}}{\sqrt {-y^{\frac {2}{5}}+a^{\frac {2}{5}}}}\right ) = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\frac {2}{5}}+{y^{\prime }}^{\frac {2}{5}}=a^{\frac {2}{5}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\left (-y^{\frac {2}{5}}+a^{\frac {2}{5}}\right )^{\frac {5}{2}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\left (-y^{\frac {2}{5}}+a^{\frac {2}{5}}\right )^{\frac {5}{2}}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\left (-y^{\frac {2}{5}}+a^{\frac {2}{5}}\right )^{\frac {5}{2}}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {5 y^{\frac {3}{5}}}{3 \left (-y^{\frac {2}{5}}+a^{\frac {2}{5}}\right )^{\frac {3}{2}}}-\frac {5 y^{\frac {1}{5}}}{\sqrt {-y^{\frac {2}{5}}+a^{\frac {2}{5}}}}+5 \arctan \left (\frac {y^{\frac {1}{5}}}{\sqrt {-y^{\frac {2}{5}}+a^{\frac {2}{5}}}}\right )=x +c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables <- differential order: 1; missing x successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 26
dsolve(y(x)^(2/5)+diff(y(x),x)^(2/5)=a^(2/5),y(x), singsol=all)
\[ x -\left (\int _{}^{y \left (x \right )}\frac {1}{\left (a^{\frac {2}{5}}-\textit {\_a}^{\frac {2}{5}}\right )^{\frac {5}{2}}}d \textit {\_a} \right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.746 (sec). Leaf size: 89
DSolve[y[x]^(2/5)+y'[x]^(2/5)==a^(2/5),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [5 \arctan \left (\frac {\sqrt [5]{\text {$\#$1}}}{\sqrt {a^{2/5}-\text {$\#$1}^{2/5}}}\right )+\frac {5 \sqrt [5]{\text {$\#$1}} \left (4 \text {$\#$1}^{2/5}-3 a^{2/5}\right )}{3 \left (a^{2/5}-\text {$\#$1}^{2/5}\right )^{3/2}}\&\right ][x+c_1] \\ y(x)\to a \\ \end{align*}