Internal problem ID [3322]
Internal file name [OUTPUT/2814_Sunday_June_05_2022_08_41_01_AM_44552190/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 3
Problem number: 58.
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, _special]]
\[ \boxed {y^{\prime }-b y^{2}=a \,x^{2}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= a \,x^{2}+b \,y^{2} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = a \,x^{2}+b \,y^{2} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=a \,x^{2}\), \(f_1(x)=0\) and \(f_2(x)=b\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{b 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' &=0\\ f_1 f_2 &=0\\ f_2^2 f_0 &=b^{2} a \,x^{2} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} b u^{\prime \prime }\left (x \right )+b^{2} a \,x^{2} u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \left (\operatorname {BesselJ}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{2} \right ) \sqrt {x} \] The above shows that \[ u^{\prime }\left (x \right ) = \sqrt {a b}\, x^{\frac {3}{2}} \left (\operatorname {BesselY}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{2} +\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{1} \right ) \] Using the above in (1) gives the solution \[ y = -\frac {\sqrt {a b}\, x \left (\operatorname {BesselY}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{2} +\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{1} \right )}{b \left (\operatorname {BesselJ}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{2} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\sqrt {a b}\, x \left (\operatorname {BesselY}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )+\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{3} \right )}{b \left (\operatorname {BesselJ}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{3} +\operatorname {BesselY}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\sqrt {a b}\, x \left (\operatorname {BesselY}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )+\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{3} \right )}{b \left (\operatorname {BesselJ}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{3} +\operatorname {BesselY}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\sqrt {a b}\, x \left (\operatorname {BesselY}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )+\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{3} \right )}{b \left (\operatorname {BesselJ}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{3} +\operatorname {BesselY}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-b y^{2}=a \,x^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=a \,x^{2}+b y^{2} \end {array} \]
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 Special <- Riccati Special successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 73
dsolve(diff(y(x),x) = a*x^2+b*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{1} -\operatorname {BesselY}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )\right ) \sqrt {a b}\, x}{b \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )+\operatorname {BesselY}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.143 (sec). Leaf size: 305
DSolve[y'[x]==a x^2+b y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a} \sqrt {b} x^2 \left (-2 \operatorname {BesselJ}\left (-\frac {3}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )+c_1 \left (\operatorname {BesselJ}\left (\frac {3}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )-\operatorname {BesselJ}\left (-\frac {5}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )\right )\right )-c_1 \operatorname {BesselJ}\left (-\frac {1}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )}{2 b x \left (\operatorname {BesselJ}\left (\frac {1}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )\right )} \\ y(x)\to -\frac {\sqrt {a} \sqrt {b} x^2 \operatorname {BesselJ}\left (-\frac {5}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )-\sqrt {a} \sqrt {b} x^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )+\operatorname {BesselJ}\left (-\frac {1}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )}{2 b x \operatorname {BesselJ}\left (-\frac {1}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )} \\ \end{align*}