2.4.32 Problem 28

Internal problem ID [8921]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 28
Date solved : Sunday, March 30, 2025 at 01:53:27 PM
CAS classification : [`y=_G(x,y')`]

Solve

$$\begin{array}{rl}{y^{\prime }}^2+y^2& =\sec {\left(x\right)}^4\end{array}$$

Solving for the derivative gives these ODE’s to solve

$$\begin{array}{}\text{(1)}& y^{\prime }& =\frac{\sqrt{1-y^2\cos {\left(x\right)}^4}}{\cos {\left(x\right)}^2}\text{(2)}& y^{\prime }& =-\frac{\sqrt{1-y^2\cos {\left(x\right)}^4}}{\cos {\left(x\right)}^2}\end{array}$$

Now each of the above is solved separately.

Solving Eq. (1)

Unknown ode type.

Solving Eq. (2)

Unknown ode type.

Unknown ode type.

✗ Maple

ode:=diff(y(x),x)^2+y(x)^2 = sec(x)^4; 
dsolve(ode,y(x), singsol=all);
 

$$\text{No solution found}$$

Maple trace

Methods for first order ODEs: 
-> Solving 1st order ODE of high degree, 1st attempt 
trying 1st order WeierstrassP solution for high degree ODE 
trying 1st order WeierstrassPPrime solution for high degree ODE 
trying 1st order JacobiSN solution for high degree ODE 
trying 1st order ODE linearizable_by_differentiation 
trying differential order: 1; missing variables 
trying simple symmetries for implicit equations 
Successful isolation of dy/dx: 2 solutions were found. Trying to solve each res\ 
ulting ODE. 
   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying homogeneous types: 
   trying exact 
   Looking for potential symmetries 
   trying an equivalence to an Abel ODE 
   trying 1st order ODE linearizable_by_differentiation 
-> Solving 1st order ODE of high degree, Lie methods, 1st trial 
   -> Computing symmetries using: way = 3 
   -> Computing symmetries using: way = 4 
   -> Computing symmetries using: way = 2 
   -> Computing symmetries using: way = 2 
-> Solving 1st order ODE of high degree, 2nd attempt. Trying parametric methods 
trying dAlembert 
-> Calling odsolve with the ODE, diff(y(x),x) = x^2/(2*(x^2+y(x)^2)^(5/4)*(((x^ 
2+y(x)^2)^(1/2)-1)/(x^2+y(x)^2)^(1/2))^(1/2)-y(x)*x), y(x) 
   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying homogeneous types: 
   trying exact 
   Looking for potential symmetries 
   trying an equivalence to an Abel ODE 
   trying 1st order ODE linearizable_by_differentiation 
-> Calling odsolve with the ODE, diff(y(x),x) = -x*(10+15*cos(2*y(x))+cos(6*y(x 
))+6*cos(4*y(x)))/(x*(16-6*x^2-2*x^2*(cos(4*y(x))+4*cos(2*y(x))))^(1/2)*cos(4*y 
(x))+4*(16-6*x^2-2*x^2*(cos(4*y(x))+4*cos(2*y(x))))^(1/2)*cos(2*y(x))*x+3*x*(16 
-6*x^2-2*x^2*(cos(4*y(x))+4*cos(2*y(x))))^(1/2)-32*sin(2*y(x))), y(x) 
   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying homogeneous types: 
   trying exact 
   Looking for potential symmetries 
   trying an equivalence to an Abel ODE 
   trying 1st order ODE linearizable_by_differentiation 
-> Solving 1st order ODE of high degree, Lie methods, 2nd trial 
   -> Computing symmetries using: way = 4 
   -> Computing symmetries using: way = 5 
   -> Computing symmetries using: way = 5
 

Maple step by step

$$\begin{array}{lll}& & \text{Let’s solve}\\ & & {\left(\frac{d}{dx}y\left(x\right)\right)}^2+y{\left(x\right)}^2=\sec {\left(x\right)}^4\\ \bullet & & \text{Highest derivative means the order of the ODE is}1\\ & & \frac{d}{dx}y\left(x\right)\\ \bullet & & \text{Solve for the highest derivative}\\ & & [\frac{d}{dx}y\left(x\right)=\frac{\sqrt{1-y{\left(x\right)}^2\cos {\left(x\right)}^4}}{\cos {\left(x\right)}^2},\frac{d}{dx}y\left(x\right)=-\frac{\sqrt{1-y{\left(x\right)}^2\cos {\left(x\right)}^4}}{\cos {\left(x\right)}^2}]\\ \bullet & & \text{Solve the equation}\frac{d}{dx}y\left(x\right)=\frac{\sqrt{1-y{\left(x\right)}^2\cos {\left(x\right)}^4}}{\cos {\left(x\right)}^2}\\ \bullet & & \text{Solve the equation}\frac{d}{dx}y\left(x\right)=-\frac{\sqrt{1-y{\left(x\right)}^2\cos {\left(x\right)}^4}}{\cos {\left(x\right)}^2}\\ \bullet & & \text{Set of solutions}\\ & & \{\mathit{workingODE},\mathit{workingODE}\}\end{array}$$

✗ Mathematica

ode=D[y[x],x]^2+y[x]^2==Sec[x]^4; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)**2 + Derivative(y(x), x)**2 - 1/cos(x)**4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-y(x)**2*cos(x)**4 + 1)/cos(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method