Internal problem ID [13348]
Internal file name [OUTPUT/12520_Wednesday_February_14_2024_11_54_36_PM_44498961/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 5. LINEAR FIRST ORDER EQUATIONS. Additional exercises. page
103
Problem number: 5.1 (c).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[_Riccati]
\[ \boxed {y^{\prime }-x y^{2}=\sqrt {x}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= y^{2} x +\sqrt {x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y^{2} x +\sqrt {x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\sqrt {x}\), \(f_1(x)=0\) and \(f_2(x)=x\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{x u} \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' &=1\\ f_1 f_2 &=0\\ f_2^2 f_0 &=x^{\frac {5}{2}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} x u^{\prime \prime }\left (x \right )-u^{\prime }\left (x \right )+x^{\frac {5}{2}} u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = x \left (c_{1} \operatorname {BesselJ}\left (\frac {4}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )+c_{2} \operatorname {BesselY}\left (\frac {4}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = x^{\frac {7}{4}} \left (\operatorname {BesselJ}\left (-\frac {3}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right ) c_{1} +\operatorname {BesselY}\left (-\frac {3}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right ) c_{2} \right ) \] Using the above in (1) gives the solution \[ y = -\frac {\operatorname {BesselJ}\left (-\frac {3}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right ) c_{1} +\operatorname {BesselY}\left (-\frac {3}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right ) c_{2}}{x^{\frac {1}{4}} \left (c_{1} \operatorname {BesselJ}\left (\frac {4}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )+c_{2} \operatorname {BesselY}\left (\frac {4}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\operatorname {BesselJ}\left (-\frac {3}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right ) c_{3} +\operatorname {BesselY}\left (-\frac {3}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )}{x^{\frac {1}{4}} \left (c_{3} \operatorname {BesselJ}\left (\frac {4}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )+\operatorname {BesselY}\left (\frac {4}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\operatorname {BesselJ}\left (-\frac {3}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right ) c_{3} +\operatorname {BesselY}\left (-\frac {3}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )}{x^{\frac {1}{4}} \left (c_{3} \operatorname {BesselJ}\left (\frac {4}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )+\operatorname {BesselY}\left (\frac {4}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\operatorname {BesselJ}\left (-\frac {3}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right ) c_{3} +\operatorname {BesselY}\left (-\frac {3}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )}{x^{\frac {1}{4}} \left (c_{3} \operatorname {BesselJ}\left (\frac {4}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )+\operatorname {BesselY}\left (\frac {4}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-x y^{2}=\sqrt {x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=x y^{2}+\sqrt {x} \end {array} \]
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 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) = (diff(y(x), x))/x-y(x)*x^(3/2), 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 <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 45
dsolve(diff(y(x),x)-x*y(x)^2=sqrt(x),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\operatorname {BesselY}\left (-\frac {3}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {3}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )}{x^{\frac {1}{4}} \left (\operatorname {BesselY}\left (\frac {4}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {4}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.242 (sec). Leaf size: 273
DSolve[y'[x]-x*y[x]^2==Sqrt[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x^{7/4} \operatorname {Gamma}\left (\frac {11}{7}\right ) \operatorname {BesselJ}\left (-\frac {3}{7},\frac {4 x^{7/4}}{7}\right )-x^{7/4} \operatorname {Gamma}\left (\frac {11}{7}\right ) \operatorname {BesselJ}\left (\frac {11}{7},\frac {4 x^{7/4}}{7}\right )+2 \operatorname {Gamma}\left (\frac {11}{7}\right ) \operatorname {BesselJ}\left (\frac {4}{7},\frac {4 x^{7/4}}{7}\right )+c_1 x^{7/4} \operatorname {Gamma}\left (\frac {3}{7}\right ) \operatorname {BesselJ}\left (-\frac {11}{7},\frac {4 x^{7/4}}{7}\right )-c_1 x^{7/4} \operatorname {Gamma}\left (\frac {3}{7}\right ) \operatorname {BesselJ}\left (\frac {3}{7},\frac {4 x^{7/4}}{7}\right )+2 c_1 \operatorname {Gamma}\left (\frac {3}{7}\right ) \operatorname {BesselJ}\left (-\frac {4}{7},\frac {4 x^{7/4}}{7}\right )}{2 x^2 \left (\operatorname {Gamma}\left (\frac {11}{7}\right ) \operatorname {BesselJ}\left (\frac {4}{7},\frac {4 x^{7/4}}{7}\right )+c_1 \operatorname {Gamma}\left (\frac {3}{7}\right ) \operatorname {BesselJ}\left (-\frac {4}{7},\frac {4 x^{7/4}}{7}\right )\right )} \\ y(x)\to -\frac {x^{7/4} \operatorname {BesselJ}\left (-\frac {11}{7},\frac {4 x^{7/4}}{7}\right )-x^{7/4} \operatorname {BesselJ}\left (\frac {3}{7},\frac {4 x^{7/4}}{7}\right )+2 \operatorname {BesselJ}\left (-\frac {4}{7},\frac {4 x^{7/4}}{7}\right )}{2 x^2 \operatorname {BesselJ}\left (-\frac {4}{7},\frac {4 x^{7/4}}{7}\right )} \\ \end{align*}