Internal problem ID [8350]
Internal file name [OUTPUT/7283_Sunday_June_05_2022_05_42_22_PM_95633877/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 13.
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]
\[ \boxed {y^{\prime }+y^{2}=a x +b} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= a x -y^{2}+b \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = a x -y^{2}+b \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=a x +b\), \(f_1(x)=0\) and \(f_2(x)=-1\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-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 &=a x +b \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -u^{\prime \prime }\left (x \right )+\left (a x +b \right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{1} \operatorname {AiryAi}\left (\frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right )+c_{2} \operatorname {AiryBi}\left (\frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \left (-\operatorname {AiryBi}\left (1, \frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right ) c_{2} -\operatorname {AiryAi}\left (1, \frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right ) c_{1} \right ) \left (-a \right )^{\frac {1}{3}} \] Using the above in (1) gives the solution \[ y = \frac {\left (-\operatorname {AiryBi}\left (1, \frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right ) c_{2} -\operatorname {AiryAi}\left (1, \frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right ) c_{1} \right ) \left (-a \right )^{\frac {1}{3}}}{c_{1} \operatorname {AiryAi}\left (\frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right )+c_{2} \operatorname {AiryBi}\left (\frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\left (-\operatorname {AiryBi}\left (1, \frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right )-\operatorname {AiryAi}\left (1, \frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right ) c_{3} \right ) \left (-a \right )^{\frac {1}{3}}}{c_{3} \operatorname {AiryAi}\left (\frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right )+\operatorname {AiryBi}\left (\frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (-\operatorname {AiryBi}\left (1, \frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right )-\operatorname {AiryAi}\left (1, \frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right ) c_{3} \right ) \left (-a \right )^{\frac {1}{3}}}{c_{3} \operatorname {AiryAi}\left (\frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right )+\operatorname {AiryBi}\left (\frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (-\operatorname {AiryBi}\left (1, \frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right )-\operatorname {AiryAi}\left (1, \frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right ) c_{3} \right ) \left (-a \right )^{\frac {1}{3}}}{c_{3} \operatorname {AiryAi}\left (\frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right )+\operatorname {AiryBi}\left (\frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y^{2}=a x +b \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y^{2}+a x +b \end {array} \]
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: 0.016 (sec). Leaf size: 79
dsolve(diff(y(x),x) + y(x)^2 - a*x - b=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {i \left (-i a \right )^{\frac {1}{3}} \left (\operatorname {AiryAi}\left (1, -\frac {a x +b}{\left (-i a \right )^{\frac {2}{3}}}\right ) c_{1} +\operatorname {AiryBi}\left (1, -\frac {a x +b}{\left (-i a \right )^{\frac {2}{3}}}\right )\right )}{\operatorname {AiryAi}\left (-\frac {a x +b}{\left (-i a \right )^{\frac {2}{3}}}\right ) c_{1} +\operatorname {AiryBi}\left (-\frac {a x +b}{\left (-i a \right )^{\frac {2}{3}}}\right )} \]
✓ Solution by Mathematica
Time used: 0.185 (sec). Leaf size: 105
DSolve[y'[x] + y[x]^2 - a*x - b==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{a} \left (\operatorname {AiryBiPrime}\left (\frac {b+a x}{a^{2/3}}\right )+c_1 \operatorname {AiryAiPrime}\left (\frac {b+a x}{a^{2/3}}\right )\right )}{\operatorname {AiryBi}\left (\frac {b+a x}{a^{2/3}}\right )+c_1 \operatorname {AiryAi}\left (\frac {b+a x}{a^{2/3}}\right )} \\ y(x)\to \frac {\sqrt [3]{a} \operatorname {AiryAiPrime}\left (\frac {b+a x}{a^{2/3}}\right )}{\operatorname {AiryAi}\left (\frac {b+a x}{a^{2/3}}\right )} \\ \end{align*}