Internal problem ID [8355]
Internal file name [OUTPUT/7288_Sunday_June_05_2022_05_42_43_PM_20896864/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 18.
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]
\[ \boxed {y^{\prime }-y^{2}-y x=x -1} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= x y +y^{2}+x -1 \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = x y +y^{2}+x -1 \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=x -1\), \(f_1(x)=x\) 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 &=x\\ f_2^2 f_0 &=x -1 \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} u^{\prime \prime }\left (x \right )-x u^{\prime }\left (x \right )+\left (x -1\right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = {\mathrm e}^{x} \left (c_{1} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -2\right )}{2}\right )+c_{2} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\left (i c_{1} {\mathrm e}^{\frac {\left (x -2\right )^{2}}{2}} \sqrt {2}+\left (c_{1} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -2\right )}{2}\right )+c_{2} \right ) \sqrt {\pi }\right ) {\mathrm e}^{x}}{\sqrt {\pi }} \] Using the above in (1) gives the solution \[ y = -\frac {i c_{1} {\mathrm e}^{\frac {\left (x -2\right )^{2}}{2}} \sqrt {2}+\left (c_{1} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -2\right )}{2}\right )+c_{2} \right ) \sqrt {\pi }}{\sqrt {\pi }\, \left (c_{1} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -2\right )}{2}\right )+c_{2} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {i c_{3} {\mathrm e}^{\frac {\left (x -2\right )^{2}}{2}} \sqrt {2}+\left (c_{3} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -2\right )}{2}\right )+1\right ) \sqrt {\pi }}{\sqrt {\pi }\, \left (c_{3} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -2\right )}{2}\right )+1\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {i c_{3} {\mathrm e}^{\frac {\left (x -2\right )^{2}}{2}} \sqrt {2}+\left (c_{3} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -2\right )}{2}\right )+1\right ) \sqrt {\pi }}{\sqrt {\pi }\, \left (c_{3} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -2\right )}{2}\right )+1\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {i c_{3} {\mathrm e}^{\frac {\left (x -2\right )^{2}}{2}} \sqrt {2}+\left (c_{3} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -2\right )}{2}\right )+1\right ) \sqrt {\pi }}{\sqrt {\pi }\, \left (c_{3} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -2\right )}{2}\right )+1\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}-y x =x -1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}+y x +x -1 \end {array} \]
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 sub-methods: <- Riccati particular polynomial solution successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 66
dsolve(diff(y(x),x) - y(x)^2 - x*y(x) - x + 1=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-i \sqrt {\pi }\, {\mathrm e}^{-2} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -2\right )}{2}\right )+2 \,{\mathrm e}^{\frac {x \left (x -4\right )}{2}}-2 c_{1}}{i \sqrt {\pi }\, {\mathrm e}^{-2} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -2\right )}{2}\right )+2 c_{1}} \]
✓ Solution by Mathematica
Time used: 0.176 (sec). Leaf size: 54
DSolve[y'[x]- y[x]^2 - x*y[x] - x + 1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -1+\frac {2 e^{\frac {1}{2} (x-2)^2}}{-\sqrt {2 \pi } \text {erfi}\left (\frac {x-2}{\sqrt {2}}\right )+2 e^2 c_1} \\ y(x)\to -1 \\ \end{align*}