2.2.12 Problem 13

Internal problem ID [9135]
Book : Second order enumerated odes
Section : section 2
Problem number : 13
Date solved : Sunday, March 30, 2025 at 02:23:06 PM
CAS classification : [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

Solved as second nonlinear ode solved by Mainardi Lioville method

Time used: 0.219 (sec)

Solve

\begin{align*} 10 y^{\prime \prime }+x^{2} y^{\prime }+\frac {3 {y^{\prime }}^{2}}{y}&=0 \end{align*}

The ode has the Liouville form given by

\begin{align*} y^{\prime \prime }+ f(x) y^{\prime } + g(y) {y^{\prime }}^{2} &= 0 \tag {1A} \end{align*}

Where in this problem

\begin{align*} f(x) &= \frac {x^{2}}{10}\\ g(y) &= \frac {3}{10 y} \end{align*}

Dividing through by \(y^{\prime }\) then Eq (1A) becomes

\begin{align*} \frac {y^{\prime \prime }}{y^{\prime }}+ f + g y^{\prime } &= 0 \tag {2A} \end{align*}

But the first term in Eq (2A) can be written as

\begin{align*} \frac {y^{\prime \prime }}{y^{\prime }}&= \frac {d}{dx} \ln \left ( y^{\prime } \right )\tag {3A} \end{align*}

And the last term in Eq (2A) can be written as

\begin{align*} g \frac {dy}{dx}&= \left ( \frac {d}{dy} \int g d y\right ) \frac {dy}{dx} \\ &= \frac {d}{dx} \int g d y\tag {4A} \end{align*}

Substituting (3A,4A) back into (2A) gives

\begin{align*} \frac {d}{dx} \ln \left ( y^{\prime } \right ) + \frac {d}{dx} \int g d y &= -f \tag {5A} \end{align*}

Integrating the above w.r.t. \(x\) gives

\begin{align*} \ln \left ( y^{\prime } \right ) + \int g d y &= - \int f d x + c_1 \end{align*}

Where \(c_1\) is arbitrary constant. Taking the exponential of the above gives

\begin{align*} y^{\prime } &= c_2 e^{\int -g d y}\, e^{\int -f d x}\tag {6A} \end{align*}

Where \(c_2\) is a new arbitrary constant. But since \(g=\frac {3}{10 y}\) and \(f=\frac {x^{2}}{10}\), then

\begin{align*} \int -g d y &= \int -\frac {3}{10 y}d y\\ &= -\frac {3 \ln \left (y\right )}{10}\\ \int -f d x &= \int -\frac {x^{2}}{10}d x\\ &= -\frac {x^{3}}{30} \end{align*}

Substituting the above into Eq(6A) gives

\[ y^{\prime } = \frac {c_2 \,{\mathrm e}^{-\frac {x^{3}}{30}}}{y^{{3}/{10}}} \]

Which is now solved as first order separable ode. The ode

\begin{equation} y^{\prime } = \frac {c_2 \,{\mathrm e}^{-\frac {x^{3}}{30}}}{y^{{3}/{10}}} \end{equation}

is separable as it can be written as

\begin{align*} y^{\prime }&= \frac {c_2 \,{\mathrm e}^{-\frac {x^{3}}{30}}}{y^{{3}/{10}}}\\ &= f(x) g(y) \end{align*}

Where

\begin{align*} f(x) &= c_2 \,{\mathrm e}^{-\frac {x^{3}}{30}}\\ g(y) &= \frac {1}{y^{{3}/{10}}} \end{align*}

Integrating gives

\begin{align*} \int { \frac {1}{g(y)} \,dy} &= \int { f(x) \,dx} \\ \int { y^{{3}/{10}}\,dy} &= \int { c_2 \,{\mathrm e}^{-\frac {x^{3}}{30}} \,dx} \\ \end{align*}

\[ \frac {10 y^{{13}/{10}}}{13}=\int c_2 \,{\mathrm e}^{-\frac {x^{3}}{30}}d x +2 c_3 \]

Will add steps showing solving for IC soon.

Summary of solutions found

\begin{align*} \frac {10 y^{{13}/{10}}}{13} &= \int c_2 \,{\mathrm e}^{-\frac {x^{3}}{30}}d x +2 c_3 \\ \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 49

ode:=10*diff(diff(y(x),x),x)+x^2*diff(y(x),x)+3/y(x)*diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {10 y^{{13}/{10}}}{13}-\frac {3 c_1 x \operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, \frac {x^{3}}{30}\right ) {\mathrm e}^{-\frac {x^{3}}{60}} 30^{{1}/{6}}}{4 \left (x^{3}\right )^{{1}/{6}}}-c_1 x \,{\mathrm e}^{-\frac {x^{3}}{30}}-c_2 = 0 \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
<- 2nd_order Liouville successful
 

✓ Mathematica. Time used: 66.264 (sec). Leaf size: 73

ode=10*D[y[x],{x,2}]+x^2*D[y[x],x]+3/y[x]*(D[y[x],x])^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2 \exp \left (\int _1^x\frac {30 e^{-\frac {1}{30} K[1]^3} \sqrt [3]{K[1]^3}}{30 c_1 \sqrt [3]{K[1]^3}-13 \sqrt [3]{30} \Gamma \left (\frac {1}{3},\frac {K[1]^3}{30}\right ) K[1]}dK[1]\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x) + 10*Derivative(y(x), (x, 2)) + 3*Derivative(y(x), x)**2/y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x**2*y(x)/6 - sqrt(-(-x**4*y(x) + 120*Derivative(y(x), (x, 2)))*y(x))/6 + Derivative(y(x), x) cannot be solved by the factorable group method