Internal problem ID [14935]
Internal file name [OUTPUT/14945_Monday_April_15_2024_12_04_06_AM_23389732/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 1. Basic concepts and definitions. Exercises page 18
Problem number: 4.
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]
\[ \boxed {y^{\prime }-y-3 y^{\frac {1}{3}}=0} \]
Integrating both sides gives \begin{align*} \int \frac {1}{y +3 y^{\frac {1}{3}}}d y &= \int d x \\ \frac {3 \ln \left (y^{\frac {2}{3}}+3\right )}{2}&=x +c_{1} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} \frac {3 \ln \left (y^{\frac {2}{3}}+3\right )}{2} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \frac {3 \ln \left (y^{\frac {2}{3}}+3\right )}{2} = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y-3 y^{\frac {1}{3}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y+3 y^{\frac {1}{3}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y+3 y^{\frac {1}{3}}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y+3 y^{\frac {1}{3}}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (y^{2}+27\right )}{2}+\ln \left (y^{\frac {2}{3}}+3\right )-\frac {\ln \left (y^{\frac {4}{3}}-3 y^{\frac {2}{3}}+9\right )}{2}=x +c_{1} \end {array} \]
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: 0.016 (sec). Leaf size: 16
dsolve(diff(y(x),x)=y(x)+3*y(x)^(1/3),y(x), singsol=all)
\[ 3+y \left (x \right )^{\frac {2}{3}}-{\mathrm e}^{\frac {2 x}{3}} c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 2.285 (sec). Leaf size: 39
DSolve[y'[x]==y[x]+3*y[x]^(1/3),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (-3+e^{\frac {2 (x+c_1)}{3}}\right ){}^{3/2} \\ y(x)\to 0 \\ y(x)\to -3 i \sqrt {3} \\ \end{align*}