Internal problem ID [3320]
Internal file name [OUTPUT/2812_Sunday_June_05_2022_08_40_55_AM_68784062/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 3
Problem number: 56.
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]
\[ \boxed {y^{\prime }-c y^{2}=b x +a} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= c \,y^{2}+b x +a \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = c \,y^{2}+b x +a \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=b x +a\), \(f_1(x)=0\) and \(f_2(x)=c\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{c 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 &=c^{2} \left (b x +a \right ) \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} c u^{\prime \prime }\left (x \right )+c^{2} \left (b x +a \right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{1} \operatorname {AiryAi}\left (-\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right )+c_{2} \operatorname {AiryBi}\left (-\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \left (-\operatorname {AiryAi}\left (1, -\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right ) c_{1} -\operatorname {AiryBi}\left (1, -\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right ) c_{2} \right ) \left (b c \right )^{\frac {1}{3}} \] Using the above in (1) gives the solution \[ y = -\frac {\left (-\operatorname {AiryAi}\left (1, -\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right ) c_{1} -\operatorname {AiryBi}\left (1, -\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right ) c_{2} \right ) \left (b c \right )^{\frac {1}{3}}}{c \left (c_{1} \operatorname {AiryAi}\left (-\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right )+c_{2} \operatorname {AiryBi}\left (-\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\left (\operatorname {AiryAi}\left (1, -\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right ) c_{3} +\operatorname {AiryBi}\left (1, -\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right )\right ) \left (b c \right )^{\frac {1}{3}}}{c \left (c_{3} \operatorname {AiryAi}\left (-\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right )+\operatorname {AiryBi}\left (-\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (\operatorname {AiryAi}\left (1, -\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right ) c_{3} +\operatorname {AiryBi}\left (1, -\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right )\right ) \left (b c \right )^{\frac {1}{3}}}{c \left (c_{3} \operatorname {AiryAi}\left (-\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right )+\operatorname {AiryBi}\left (-\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (\operatorname {AiryAi}\left (1, -\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right ) c_{3} +\operatorname {AiryBi}\left (1, -\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right )\right ) \left (b c \right )^{\frac {1}{3}}}{c \left (c_{3} \operatorname {AiryAi}\left (-\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right )+\operatorname {AiryBi}\left (-\frac {\left (b c \right )^{\frac {1}{3}} \left (b x +a \right )}{b}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-c y^{2}=b x +a \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=a +b x +c y^{2} \end {array} \]
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 trying Riccati sub-methods: <- Abel AIR successful: ODE belongs to the 0F1 0-parameter (Airy type) class`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 85
dsolve(diff(y(x),x) = a+b*x+c*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\frac {b}{\sqrt {c}}\right )^{\frac {1}{3}} \left (\operatorname {AiryAi}\left (1, -\frac {b x +a}{\left (\frac {b}{\sqrt {c}}\right )^{\frac {2}{3}}}\right ) c_{1} +\operatorname {AiryBi}\left (1, -\frac {b x +a}{\left (\frac {b}{\sqrt {c}}\right )^{\frac {2}{3}}}\right )\right )}{\sqrt {c}\, \left (c_{1} \operatorname {AiryAi}\left (-\frac {b x +a}{\left (\frac {b}{\sqrt {c}}\right )^{\frac {2}{3}}}\right )+\operatorname {AiryBi}\left (-\frac {b x +a}{\left (\frac {b}{\sqrt {c}}\right )^{\frac {2}{3}}}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.199 (sec). Leaf size: 143
DSolve[y'[x]==a+b x+c y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {b \left (\operatorname {AiryBiPrime}\left (-\frac {c (a+b x)}{(-b c)^{2/3}}\right )+c_1 \operatorname {AiryAiPrime}\left (-\frac {c (a+b x)}{(-b c)^{2/3}}\right )\right )}{(-b c)^{2/3} \left (\operatorname {AiryBi}\left (-\frac {c (a+b x)}{(-b c)^{2/3}}\right )+c_1 \operatorname {AiryAi}\left (-\frac {c (a+b x)}{(-b c)^{2/3}}\right )\right )} \\ y(x)\to \frac {b \operatorname {AiryAiPrime}\left (-\frac {c (a+b x)}{(-b c)^{2/3}}\right )}{(-b c)^{2/3} \operatorname {AiryAi}\left (-\frac {c (a+b x)}{(-b c)^{2/3}}\right )} \\ \end{align*}