Internal problem ID [15125]
Internal file name [OUTPUT/15126_Sunday_April_21_2024_01_34_29_PM_73397113/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 11. Singular solutions of differential equations. Exercises page 92
Problem number: 264.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y^{\prime }-y^{\frac {2}{3}}=a} \]
Integrating both sides gives \begin{align*} \int \frac {1}{y^{\frac {2}{3}}+a}d y &= \int d x \\ 3 y^{\frac {1}{3}}-3 \sqrt {a}\, \arctan \left (\frac {y^{\frac {1}{3}}}{\sqrt {a}}\right )&=x +c_{1} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} 3 y^{\frac {1}{3}}-3 \sqrt {a}\, \arctan \left (\frac {y^{\frac {1}{3}}}{\sqrt {a}}\right ) &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ 3 y^{\frac {1}{3}}-3 \sqrt {a}\, \arctan \left (\frac {y^{\frac {1}{3}}}{\sqrt {a}}\right ) = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{\frac {2}{3}}=a \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{\frac {2}{3}}+a \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{\frac {2}{3}}+a}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{\frac {2}{3}}+a}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & 3 y^{\frac {1}{3}}-\sqrt {a}\, \arctan \left (\frac {2 y^{\frac {1}{3}}+\sqrt {3}\, \sqrt {a}}{\sqrt {a}}\right )+\sqrt {a}\, \arctan \left (\frac {\sqrt {3}\, \sqrt {a}-2 y^{\frac {1}{3}}}{\sqrt {a}}\right )-2 \sqrt {a}\, \arctan \left (\frac {y^{\frac {1}{3}}}{\sqrt {a}}\right )+\sqrt {a}\, \arctan \left (\frac {y}{a^{\frac {3}{2}}}\right )=x +c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 85
dsolve(diff(y(x),x)=y(x)^(2/3)+a,y(x), singsol=all)
\[ x -3 y \left (x \right )^{\frac {1}{3}}+2 \sqrt {a}\, \arctan \left (\frac {y \left (x \right )^{\frac {1}{3}}}{\sqrt {a}}\right )-\sqrt {a}\, \arctan \left (\frac {\sqrt {3}\, \sqrt {a}-2 y \left (x \right )^{\frac {1}{3}}}{\sqrt {a}}\right )+\sqrt {a}\, \arctan \left (\frac {2 y \left (x \right )^{\frac {1}{3}}+\sqrt {3}\, \sqrt {a}}{\sqrt {a}}\right )-\sqrt {a}\, \arctan \left (\frac {y \left (x \right )}{a^{\frac {3}{2}}}\right )+c_{1} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]=y[x]^(2/3)+a,y[x],x,IncludeSingularSolutions -> True]
Not solved