2.41 problem 41

2.41.1 Solved as first order ode of type Riccati
2.41.2 Maple step by step solution
2.41.3 Maple trace
2.41.4 Maple dsolve solution
2.41.5 Mathematica DSolve solution

Internal problem ID [8130]
Book : Second order enumerated odes
Section : section 2
Problem number : 41
Date solved : Monday, October 21, 2024 at 04:54:08 PM
CAS classification : [[_Riccati, _special]]

Solve

\begin{align*} y^{\prime }&=x -y^{2} \end{align*}

2.41.1 Solved as first order ode of type Riccati

Time used: 0.266 (sec)

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= -y^{2}+x \end{align*}

This is a Riccati ODE. Comparing the ODE to solve

\[ y' = -y^{2}+x \]

With Riccati ODE standard form

\[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]

Shows that \(f_0(x)=x\), \(f_1(x)=0\) 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 &=x \end{align*}

Substituting the above terms back in equation (2) gives

\begin{align*} -u^{\prime \prime }\left (x \right )+x u \left (x \right ) = 0 \end{align*}

This is Airy ODE. It has the general form

\[ a \frac {d^{2}u}{d x^{2}} + b \frac {d u}{d x} + c x u = F(x) \]

Where in this case

\begin{align*} a &= -1\\ b &= 0\\ c &= 1\\ F &= 0 \end{align*}

Therefore the solution to the homogeneous Airy ODE becomes

\[ u = c_1 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_2 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right ) \]

Will add steps showing solving for IC soon.

Taking derivative gives

\[ u^{\prime }\left (x \right ) = -c_1 \left (-1\right )^{{1}/{3}} \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right )-c_2 \left (-1\right )^{{1}/{3}} \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) \]

Doing change of constants, the solution becomes

\[ y = \frac {-c_3 \left (-1\right )^{{1}/{3}} \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right )-\left (-1\right )^{{1}/{3}} \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right )}{c_3 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+\operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )} \]
Figure 163: Slope field plot
\(y^{\prime } = x -y^{2}\)
2.41.2 Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=x -y^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=x -y^{2} \end {array} \]

2.41.3 Maple trace
Methods for first order ODEs:
 
2.41.4 Maple dsolve solution

Solving time : 0.003 (sec)
Leaf size : 23

dsolve(diff(y(x),x) = x-y(x)^2, 
       y(x),singsol=all)
 
\[ y = \frac {c_1 \operatorname {AiryAi}\left (1, x\right )+\operatorname {AiryBi}\left (1, x\right )}{c_1 \operatorname {AiryAi}\left (x \right )+\operatorname {AiryBi}\left (x \right )} \]
2.41.5 Mathematica DSolve solution

Solving time : 0.18 (sec)
Leaf size : 223

DSolve[{D[y[x],x]==x-y[x]^2,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 
\begin{align*} y(x)\to -\frac {-i x^{3/2} \left (2 \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2}{3} i x^{3/2}\right )+c_1 \left (\operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} i x^{3/2}\right )-\operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} i x^{3/2}\right )\right )\right )-c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i x^{3/2}\right )}{2 x \left (\operatorname {BesselJ}\left (\frac {1}{3},\frac {2}{3} i x^{3/2}\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i x^{3/2}\right )\right )} \\ y(x)\to \frac {i x^{3/2} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} i x^{3/2}\right )-i x^{3/2} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} i x^{3/2}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i x^{3/2}\right )}{2 x \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i x^{3/2}\right )} \\ \end{align*}