Internal problem ID [3502]
Internal file name [OUTPUT/2995_Sunday_June_05_2022_08_49_07_AM_18535337/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 9
Problem number: 246.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[_rational, _Riccati]
\[ \boxed {3 y^{\prime } x -\left (1-3 y\right ) y=3 x^{\frac {2}{3}}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {3 x^{\frac {2}{3}}-3 y^{2}+y}{3 x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {1}{x^{\frac {1}{3}}}-\frac {y^{2}}{x}+\frac {y}{3 x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {1}{x^{\frac {1}{3}}}\), \(f_1(x)=\frac {1}{3 x}\) and \(f_2(x)=-\frac {1}{x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-\frac {u}{x}} \tag {1} \end {align*}
Using the above substitution in the given ODE results (after some simplification)in a second order ODE to solve for \(u(x)\) which is \begin {align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end {align*}
But \begin {align*} f_2' &=\frac {1}{x^{2}}\\ f_1 f_2 &=-\frac {1}{3 x^{2}}\\ f_2^2 f_0 &=\frac {1}{x^{\frac {7}{3}}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -\frac {u^{\prime \prime }\left (x \right )}{x}-\frac {2 u^{\prime }\left (x \right )}{3 x^{2}}+\frac {u \left (x \right )}{x^{\frac {7}{3}}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{1} \sin \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )+c_{2} \cos \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {c_{2} \sin \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )-c_{1} \cos \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )}{x^{\frac {4}{3}} \sqrt {-\frac {1}{x^{\frac {4}{3}}}}} \] Using the above in (1) gives the solution \[ y = \frac {c_{2} \sin \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )-c_{1} \cos \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )}{x^{\frac {1}{3}} \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\, \left (c_{1} \sin \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )+c_{2} \cos \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\sin \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )-c_{3} \cos \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )}{x^{\frac {1}{3}} \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\, \left (c_{3} \sin \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )+\cos \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sin \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )-c_{3} \cos \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )}{x^{\frac {1}{3}} \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\, \left (c_{3} \sin \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )+\cos \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sin \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )-c_{3} \cos \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )}{x^{\frac {1}{3}} \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\, \left (c_{3} \sin \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )+\cos \left (3 x \sqrt {-\frac {1}{x^{\frac {4}{3}}}}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 3 y^{\prime } x -\left (1-3 y\right ) y=3 x^{\frac {2}{3}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {3 x^{\frac {2}{3}}+\left (1-3 y\right ) y}{3 x} \end {array} \]
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: 0.0 (sec). Leaf size: 19
dsolve(3*x*diff(y(x),x) = 3*x^(2/3)+(1-3*y(x))*y(x),y(x), singsol=all)
\[ y \left (x \right ) = i \tan \left (-3 i x^{\frac {1}{3}}+c_{1} \right ) x^{\frac {1}{3}} \]
✓ Solution by Mathematica
Time used: 0.181 (sec). Leaf size: 79
DSolve[3 x y'[x]==3 x^(2/3)+(1-3 y[x])y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{x} \left (i \cosh \left (3 \sqrt [3]{x}\right )+c_1 \sinh \left (3 \sqrt [3]{x}\right )\right )}{i \sinh \left (3 \sqrt [3]{x}\right )+c_1 \cosh \left (3 \sqrt [3]{x}\right )} \\ y(x)\to \sqrt [3]{x} \tanh \left (3 \sqrt [3]{x}\right ) \\ \end{align*}