Internal problem ID [3421]
Internal file name [OUTPUT/2914_Sunday_June_05_2022_08_46_58_AM_15277284/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 6
Problem number: 165.
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]
\[ \boxed {x y^{\prime }-y+y^{2}=x^{\frac {2}{3}}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {y -y^{2}+x^{\frac {2}{3}}}{x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {1}{x^{\frac {1}{3}}}-\frac {y^{2}}{x}+\frac {y}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {1}{x^{\frac {1}{3}}}\), \(f_1(x)=\frac {1}{x}\) 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}{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 {u \left (x \right )}{x^{\frac {7}{3}}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{2} {\mathrm e}^{3 x^{\frac {1}{3}}} \left (3 x^{\frac {1}{3}}-1\right )+3 \,{\mathrm e}^{-3 x^{\frac {1}{3}}} c_{1} \left (x^{\frac {1}{3}}+\frac {1}{3}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {3 \left (-c_{2} {\mathrm e}^{3 x^{\frac {1}{3}}}+c_{1} {\mathrm e}^{-3 x^{\frac {1}{3}}}\right )}{x^{\frac {1}{3}}} \] Using the above in (1) gives the solution \[ y = -\frac {3 \left (-c_{2} {\mathrm e}^{3 x^{\frac {1}{3}}}+c_{1} {\mathrm e}^{-3 x^{\frac {1}{3}}}\right ) x^{\frac {2}{3}}}{c_{2} {\mathrm e}^{3 x^{\frac {1}{3}}} \left (3 x^{\frac {1}{3}}-1\right )+3 \,{\mathrm e}^{-3 x^{\frac {1}{3}}} c_{1} \left (x^{\frac {1}{3}}+\frac {1}{3}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {3 x^{\frac {2}{3}} \left (-{\mathrm e}^{6 x^{\frac {1}{3}}}+c_{3} \right )}{\left (3 x^{\frac {1}{3}}-1\right ) {\mathrm e}^{6 x^{\frac {1}{3}}}+3 x^{\frac {1}{3}} c_{3} +c_{3}} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {3 x^{\frac {2}{3}} \left (-{\mathrm e}^{6 x^{\frac {1}{3}}}+c_{3} \right )}{\left (3 x^{\frac {1}{3}}-1\right ) {\mathrm e}^{6 x^{\frac {1}{3}}}+3 x^{\frac {1}{3}} c_{3} +c_{3}} \\ \end{align*}
Verification of solutions
\[ y = -\frac {3 x^{\frac {2}{3}} \left (-{\mathrm e}^{6 x^{\frac {1}{3}}}+c_{3} \right )}{\left (3 x^{\frac {1}{3}}-1\right ) {\mathrm e}^{6 x^{\frac {1}{3}}}+3 x^{\frac {1}{3}} c_{3} +c_{3}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }-y+y^{2}=x^{\frac {2}{3}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y-y^{2}+x^{\frac {2}{3}}}{x} \end {array} \]
Maple trace Kovacic algorithm successful
`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 differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying Riccati trying Riccati sub-methods: trying Riccati_symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = y(x)/x^(4/3), y(x)` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Group is reducible or imprimitive <- Kovacics algorithm successful <- Equivalence, under non-integer power transformations successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 72
dsolve(x*diff(y(x),x)-y(x)+y(x)^2 = x^(2/3),y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{\frac {1}{3}} \left (c_{1} {\mathrm e}^{6 x^{\frac {1}{3}}} \operatorname {abs}\left (1, 3 x^{\frac {1}{3}}-1\right )+c_{1} {\mathrm e}^{6 x^{\frac {1}{3}}} {| 3 x^{\frac {1}{3}}-1|}-3 x^{\frac {1}{3}}\right )}{c_{1} {\mathrm e}^{6 x^{\frac {1}{3}}} {| 3 x^{\frac {1}{3}}-1|}+3 x^{\frac {1}{3}}+1} \]
✓ Solution by Mathematica
Time used: 0.203 (sec). Leaf size: 131
DSolve[x y'[x]-y[x]+y[x]^2==x^(2/3),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {3 x^{2/3} \left (c_1 \cosh \left (3 \sqrt [3]{x}\right )-i \sinh \left (3 \sqrt [3]{x}\right )\right )}{\left (-3 i \sqrt [3]{x}-c_1\right ) \cosh \left (3 \sqrt [3]{x}\right )+\left (3 c_1 \sqrt [3]{x}+i\right ) \sinh \left (3 \sqrt [3]{x}\right )} \\ y(x)\to \frac {3 x^{2/3} \cosh \left (3 \sqrt [3]{x}\right )}{3 \sqrt [3]{x} \sinh \left (3 \sqrt [3]{x}\right )-\cosh \left (3 \sqrt [3]{x}\right )} \\ \end{align*}