Internal problem ID [8365]
Internal file name [OUTPUT/7298_Sunday_June_05_2022_05_43_32_PM_22341209/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 28.
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]
\[ \boxed {y^{\prime }+x y^{2}-x^{3} y=2 x} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= x^{3} y -x \,y^{2}+2 x \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = x^{3} y -x \,y^{2}+2 x \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=2 x\), \(f_1(x)=x^{3}\) and \(f_2(x)=-x\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-x 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' &=-1\\ f_1 f_2 &=-x^{4}\\ f_2^2 f_0 &=2 x^{3} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -x u^{\prime \prime }\left (x \right )-\left (-x^{4}-1\right ) u^{\prime }\left (x \right )+2 x^{3} u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = {\mathrm e}^{\frac {x^{4}}{4}} \left (c_{1} +\operatorname {erf}\left (\frac {x^{2}}{2}\right ) c_{2} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\left (x^{2} \sqrt {\pi }\, \left (c_{1} +\operatorname {erf}\left (\frac {x^{2}}{2}\right ) c_{2} \right ) {\mathrm e}^{\frac {x^{4}}{4}}+2 c_{2} \right ) x}{\sqrt {\pi }} \] Using the above in (1) gives the solution \[ y = \frac {\left (x^{2} \sqrt {\pi }\, \left (c_{1} +\operatorname {erf}\left (\frac {x^{2}}{2}\right ) c_{2} \right ) {\mathrm e}^{\frac {x^{4}}{4}}+2 c_{2} \right ) {\mathrm e}^{-\frac {x^{4}}{4}}}{\sqrt {\pi }\, \left (c_{1} +\operatorname {erf}\left (\frac {x^{2}}{2}\right ) c_{2} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\sqrt {\pi }\, \operatorname {erf}\left (\frac {x^{2}}{2}\right ) x^{2}+c_{3} \sqrt {\pi }\, x^{2}+2 \,{\mathrm e}^{-\frac {x^{4}}{4}}}{\sqrt {\pi }\, \left (c_{3} +\operatorname {erf}\left (\frac {x^{2}}{2}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sqrt {\pi }\, \operatorname {erf}\left (\frac {x^{2}}{2}\right ) x^{2}+c_{3} \sqrt {\pi }\, x^{2}+2 \,{\mathrm e}^{-\frac {x^{4}}{4}}}{\sqrt {\pi }\, \left (c_{3} +\operatorname {erf}\left (\frac {x^{2}}{2}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {\pi }\, \operatorname {erf}\left (\frac {x^{2}}{2}\right ) x^{2}+c_{3} \sqrt {\pi }\, x^{2}+2 \,{\mathrm e}^{-\frac {x^{4}}{4}}}{\sqrt {\pi }\, \left (c_{3} +\operatorname {erf}\left (\frac {x^{2}}{2}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+x y^{2}-x^{3} y=2 x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-x y^{2}+x^{3} y+2 x \end {array} \]
Maple trace Kovacic algorithm successful
`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: trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = (x^4+1)*(diff(y(x), x))/x+2*y(x)*x^2, y(x)` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Reducible group (found an exponential solution) Group is reducible, not completely reducible <- Kovacics algorithm successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 51
dsolve(diff(y(x),x) + x*y(x)^2 -x^3*y(x) - 2*x=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sqrt {\pi }\, \operatorname {erf}\left (\frac {x^{2}}{2}\right ) c_{1} x^{2}+\sqrt {\pi }\, x^{2}+2 \,{\mathrm e}^{-\frac {x^{4}}{4}} c_{1}}{\sqrt {\pi }\, \left (\operatorname {erf}\left (\frac {x^{2}}{2}\right ) c_{1} +1\right )} \]
✓ Solution by Mathematica
Time used: 0.317 (sec). Leaf size: 70
DSolve[y'[x] + x*y[x]^2 -x^3*y[x] - 2*x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {\pi } x^2 \text {erf}\left (\frac {x^2}{2}\right )+2 e^{-\frac {x^4}{4}}+2 c_1 x^2}{\sqrt {\pi } \text {erf}\left (\frac {x^2}{2}\right )+2 c_1} \\ y(x)\to x^2 \\ \end{align*}