Internal problem ID [13296]
Internal file name [OUTPUT/12468_Wednesday_February_14_2024_02_06_30_AM_77151273/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 3. Some basics about First order equations. Additional exercises. page
63
Problem number: 3.4 j.
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]
\[ \boxed {y^{\prime }+\left (8-x \right ) y-y^{2}=-8 x} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= y x +y^{2}-8 x -8 y \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y x +y^{2}-8 x -8 y \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-8 x\), \(f_1(x)=x -8\) 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 -8\\ f_2^2 f_0 &=-8 x \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} u^{\prime \prime }\left (x \right )-\left (x -8\right ) u^{\prime }\left (x \right )-8 x u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = {\mathrm e}^{-8 x} \left (c_{1} +\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right ) c_{2} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {{\mathrm e}^{-8 x} \left (i {\mathrm e}^{\frac {\left (x +8\right )^{2}}{2}} \sqrt {2}\, c_{2} -8 \sqrt {\pi }\, \left (c_{1} +\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right ) c_{2} \right )\right )}{\sqrt {\pi }} \] Using the above in (1) gives the solution \[ y = -\frac {i {\mathrm e}^{\frac {\left (x +8\right )^{2}}{2}} \sqrt {2}\, c_{2} -8 \sqrt {\pi }\, \left (c_{1} +\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right ) c_{2} \right )}{\sqrt {\pi }\, \left (c_{1} +\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right ) c_{2} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {i {\mathrm e}^{\frac {\left (x +8\right )^{2}}{2}} \sqrt {2}-8 \sqrt {\pi }\, \left (c_{3} +\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right )\right )}{\sqrt {\pi }\, \left (c_{3} +\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {i {\mathrm e}^{\frac {\left (x +8\right )^{2}}{2}} \sqrt {2}-8 \sqrt {\pi }\, \left (c_{3} +\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right )\right )}{\sqrt {\pi }\, \left (c_{3} +\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {i {\mathrm e}^{\frac {\left (x +8\right )^{2}}{2}} \sqrt {2}-8 \sqrt {\pi }\, \left (c_{3} +\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right )\right )}{\sqrt {\pi }\, \left (c_{3} +\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+\left (8-x \right ) y-y^{2}=-8 x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\left (8-x \right ) y+y^{2}-8 x \end {array} \]
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: 0.016 (sec). Leaf size: 66
dsolve(diff(y(x),x)+(8-x)*y(x)-y(x)^2=-8*x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {8 i \sqrt {\pi }\, {\mathrm e}^{-32} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right )+2 \,{\mathrm e}^{\frac {x \left (x +16\right )}{2}}+16 c_{1}}{i \sqrt {\pi }\, {\mathrm e}^{-32} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +8\right )}{2}\right )+2 c_{1}} \]
✓ Solution by Mathematica
Time used: 0.176 (sec). Leaf size: 54
DSolve[y'[x]+(8-x)*y[x]-y[x]^2==-8*x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 8+\frac {2 e^{\frac {1}{2} (x+8)^2}}{-\sqrt {2 \pi } \text {erfi}\left (\frac {x+8}{\sqrt {2}}\right )+2 e^{32} c_1} \\ y(x)\to 8 \\ \end{align*}