2.3.26 Problem 26

Internal problem ID [8884]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 26
Date solved : Sunday, March 30, 2025 at 01:52:16 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

Solved as second order missing x ode

Time used: 0.707 (sec)

Solve

$$\begin{array}{rl}{y}^{\prime \prime }+\sin \left(y\right){y^{\prime }}^2& =0\end{array}$$

This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable $y$ an independent variable. Using

$$\begin{array}{rl}{y}^{\prime }& =p\end{array}$$

Then

$$\begin{array}{rl}{y}^{″}& =\frac{dp}{dx}\\ & =\frac{dp}{dy}\frac{dy}{dx}\\ & =p\frac{dp}{dy}\end{array}$$

Hence the ode becomes

$$\begin{array}{r}p\left(y\right)\left(\frac{d}{dy}p\left(y\right)\right)+\sin \left(y\right)p{\left(y\right)}^2=0\end{array}$$

Which is now solved as first order ode for $p(y)$.

The ode

$$\begin{array}{}\text{(1)}& p^{\prime }=-\sin \left(y\right)p\end{array}$$

is separable as it can be written as

$$\begin{array}{rl}{p}^{\prime }& =-\sin \left(y\right)p\\ & =f(y)g(p)\end{array}$$

Where

$$\begin{array}{rl}f(y)& =-\sin \left(y\right)\\ g(p)& =p\end{array}$$

Integrating gives

$$\begin{array}{rl}\int \frac{1}{g(p)}dp& =\int f(y)dy\\ \int \frac{1}{p}dp& =\int -\sin \left(y\right)dy\end{array}$$

$$\ln \left(p\right)=\cos \left(y\right)+c_1$$

We now need to find the singular solutions, these are found by finding for what values $g(p)$ is zero, since we had to divide by this above. Solving $g(p)=0$ or

$$p=0$$

for $p$ gives

$$\begin{array}{rl}p& =0\end{array}$$

Now we go over each such singular solution and check if it verifies the ode itself and any initial conditions given. If it does not then the singular solution will not be used.

Therefore the solutions found are

$$\begin{array}{rl}\ln \left(p\right)& =\cos \left(y\right)+c_1\\ p& =0\end{array}$$

For solution (1) found earlier, since $p=y^{\prime }$ then we now have a new first order ode to solve which is

$$\begin{array}{r}\ln \left(y^{\prime }\right)=\cos \left(y\right)+c_1\end{array}$$

Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as

$${\int }_{}^y{\mathrm{e}}^{-\cos \left(\tau \right)-c_1}d\tau =x+c_2$$

For solution (2) found earlier, since $p=y^{\prime }$ then we now have a new first order ode to solve which is

$$\begin{array}{r}{y}^{\prime }=0\end{array}$$

Since the ode has the form $y^{\prime }=f(x)$, then we only need to integrate $f(x)$.

$$\begin{array}{rl}\int dy& =\int 0dx+c_3\\ y& =c_3\end{array}$$

Will add steps showing solving for IC soon.

Summary of solutions found

$$\begin{array}{rl}{\int }_{}^y{\mathrm{e}}^{-\cos \left(\tau \right)-c_1}d\tau & =x+c_2\\ y& =c_3\end{array}$$

✓ Maple. Time used: 0.003 (sec). Leaf size: 21

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

$${\int }_{}^y{\mathrm{e}}^{-\cos \left(\mathit{\text{\_a}}\right)}d\mathit{\text{\_a}}-c_{1}x-c_2=0$$

Maple trace

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

✓ Mathematica. Time used: 1.336 (sec). Leaf size: 111

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

$$\begin{array}{r}y(x)\to \text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{e^{\sin (K[1])-\cos (K[1])K[1]}}{c_1}dK[1]\mathrm{\&}][x+c_2]\\ y(x)\to \text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}-\frac{e^{\sin (K[1])-\cos (K[1])K[1]}}{c_1}dK[1]\mathrm{\&}][x+c_2]\\ y(x)\to \text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{e^{\sin (K[1])-\cos (K[1])K[1]}}{c_1}dK[1]\mathrm{\&}][x+c_2]\end{array}$$

✗ Sympy

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

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