Internal problem ID [531]
Internal file name [OUTPUT/531_Sunday_June_05_2022_01_43_09_AM_59273030/index.tex]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Section 2.4. Page 76
Problem number: 20.
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]
\[ \boxed {y^{\prime }+y^{2}=-1+t} \]
In canonical form the ODE is \begin {align*} y' &= F(t,y)\\ &= -y^{2}+t -1 \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -y^{2}+t -1 \] With Riccati ODE standard form \[ y' = f_0(t)+ f_1(t)y+f_2(t)y^{2} \] Shows that \(f_0(t)=-1+t\), \(f_1(t)=0\) and \(f_2(t)=-1\). 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''(t) -\left ( f_2' + f_1 f_2 \right ) u'(t) + f_2^2 f_0 u(t) &= 0 \tag {2} \end {align*}
But \begin {align*} f_2' &=0\\ f_1 f_2 &=0\\ f_2^2 f_0 &=-1+t \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -u^{\prime \prime }\left (t \right )+\left (-1+t \right ) u \left (t \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (t \right ) = c_{1} \operatorname {AiryAi}\left (-1+t \right )+c_{2} \operatorname {AiryBi}\left (-1+t \right ) \] The above shows that \[ u^{\prime }\left (t \right ) = c_{1} \operatorname {AiryAi}\left (1, -1+t \right )+c_{2} \operatorname {AiryBi}\left (1, -1+t \right ) \] Using the above in (1) gives the solution \[ y = \frac {c_{1} \operatorname {AiryAi}\left (1, -1+t \right )+c_{2} \operatorname {AiryBi}\left (1, -1+t \right )}{c_{1} \operatorname {AiryAi}\left (-1+t \right )+c_{2} \operatorname {AiryBi}\left (-1+t \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {c_{3} \operatorname {AiryAi}\left (1, -1+t \right )+\operatorname {AiryBi}\left (1, -1+t \right )}{c_{3} \operatorname {AiryAi}\left (-1+t \right )+\operatorname {AiryBi}\left (-1+t \right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{3} \operatorname {AiryAi}\left (1, -1+t \right )+\operatorname {AiryBi}\left (1, -1+t \right )}{c_{3} \operatorname {AiryAi}\left (-1+t \right )+\operatorname {AiryBi}\left (-1+t \right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{3} \operatorname {AiryAi}\left (1, -1+t \right )+\operatorname {AiryBi}\left (1, -1+t \right )}{c_{3} \operatorname {AiryAi}\left (-1+t \right )+\operatorname {AiryBi}\left (-1+t \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y^{2}=-1+t \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=t -1-y^{2} \end {array} \]
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: 0.0 (sec). Leaf size: 31
dsolve(diff(y(t),t) = t-1-y(t)^2,y(t), singsol=all)
\[ y \left (t \right ) = \frac {\operatorname {AiryAi}\left (1, t -1\right ) c_{1} +\operatorname {AiryBi}\left (1, t -1\right )}{\operatorname {AiryAi}\left (t -1\right ) c_{1} +\operatorname {AiryBi}\left (t -1\right )} \]
✓ Solution by Mathematica
Time used: 0.123 (sec). Leaf size: 47
DSolve[y'[t] == t-1-y[t]^2,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {\operatorname {AiryBiPrime}(t-1)+c_1 \operatorname {AiryAiPrime}(t-1)}{\operatorname {AiryBi}(t-1)+c_1 \operatorname {AiryAi}(t-1)} \\ y(t)\to \frac {\operatorname {AiryAiPrime}(t-1)}{\operatorname {AiryAi}(t-1)} \\ \end{align*}