Internal problem ID [11672]
Internal file name [OUTPUT/11682_Wednesday_April_10_2024_04_54_35_PM_23538320/index.tex]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 2, section 2.3 (Linear equations). Exercises page 56
Problem number: 40.
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]
\[ \boxed {y^{\prime }+y^{2}-x y=1} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= x y -y^{2}+1 \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = x y -y^{2}+1 \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=1\), \(f_1(x)=x\) 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 &=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 )+u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = {\mathrm e}^{\frac {x^{2}}{2}} \left (c_{1} \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )+c_{2} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {x \sqrt {\pi }\, \left (c_{1} \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )+c_{2} \right ) {\mathrm e}^{\frac {x^{2}}{2}}+c_{1} \sqrt {2}}{\sqrt {\pi }} \] Using the above in (1) gives the solution \[ y = \frac {\left (x \sqrt {\pi }\, \left (c_{1} \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )+c_{2} \right ) {\mathrm e}^{\frac {x^{2}}{2}}+c_{1} \sqrt {2}\right ) {\mathrm e}^{-\frac {x^{2}}{2}}}{\sqrt {\pi }\, \left (c_{1} \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )+c_{2} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {c_{3} \sqrt {2}\, {\mathrm e}^{-\frac {x^{2}}{2}}+x \sqrt {\pi }\, \left (c_{3} \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )+1\right )}{\sqrt {\pi }\, \left (c_{3} \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )+1\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{3} \sqrt {2}\, {\mathrm e}^{-\frac {x^{2}}{2}}+x \sqrt {\pi }\, \left (c_{3} \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )+1\right )}{\sqrt {\pi }\, \left (c_{3} \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )+1\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{3} \sqrt {2}\, {\mathrm e}^{-\frac {x^{2}}{2}}+x \sqrt {\pi }\, \left (c_{3} \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )+1\right )}{\sqrt {\pi }\, \left (c_{3} \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )+1\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y^{2}-x y=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y^{2}+x y+1 \end {array} \]
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 found: 2 potential symmetries. Proceeding with integration step`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 53
dsolve(diff(y(x),x)=-y(x)^2+x*y(x)+1,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right ) x +2 c_{1} x +2 \,{\mathrm e}^{-\frac {x^{2}}{2}}}{\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )+2 c_{1}} \]
✓ Solution by Mathematica
Time used: 0.154 (sec). Leaf size: 45
DSolve[y'[x]==-y[x]^2+x*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x+\frac {e^{-\frac {x^2}{2}}}{\sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {x}{\sqrt {2}}\right )+c_1} \\ y(x)\to x \\ \end{align*}