4.32 problem 28

Internal problem ID [7901]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 28
Date solved : Monday, October 21, 2024 at 04:31:50 PM
CAS classification : [`y=_G(x,y')`]

Solve

\begin{align*} {y^{\prime }}^{2}+y^{2}&=\sec \left (x \right )^{4} \end{align*}

Solving for the derivative gives these ODE’s to solve

\begin{align*} \tag{1} y^{\prime }&=\frac {\sqrt {1-y^{2} \cos \left (x \right )^{4}}}{\cos \left (x \right )^{2}} \\ \tag{2} y^{\prime }&=-\frac {\sqrt {1-y^{2} \cos \left (x \right )^{4}}}{\cos \left (x \right )^{2}} \\ \end{align*}

Now each of the above is solved separately.

Solving Eq. (1)

Solving Eq. (2)

4.32.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}+y^{2}=\sec \left (x \right )^{4} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {\sqrt {1-y^{2} \cos \left (x \right )^{4}}}{\cos \left (x \right )^{2}}, y^{\prime }=-\frac {\sqrt {1-y^{2} \cos \left (x \right )^{4}}}{\cos \left (x \right )^{2}}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\sqrt {1-y^{2} \cos \left (x \right )^{4}}}{\cos \left (x \right )^{2}} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\sqrt {1-y^{2} \cos \left (x \right )^{4}}}{\cos \left (x \right )^{2}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]

4.32.2 Maple trace

Methods for first order ODEs:
 

4.32.3 Maple dsolve solution

Solving time : 0.073 (sec)
Leaf size : maple_leaf_size

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

\[ \text {No solution found} \]

4.32.4 Mathematica DSolve solution

Solving time : 0.0 (sec)
Leaf size : 0

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

Not solved