2.4.72 Problem 69

Internal problem ID [10235]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 69
Date solved : Thursday, November 27, 2025 at 10:27:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

Solved as second order ode using change of variable on x method 2

Time used: 0.345 (sec)

Solve

\begin{align*} y^{\prime \prime } \sin \left (2 x \right )^{2}+y^{\prime } \sin \left (4 x \right )-4 y&=0 \\ \end{align*}

In normal form the ode

\begin{align*} y^{\prime \prime } \sin \left (2 x \right )^{2}+y^{\prime } \sin \left (4 x \right )-4 y&=0 \tag {1} \end{align*}

Becomes

\begin{align*} y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y&=0 \tag {2} \end{align*}

Where

\begin{align*} p \left (x \right )&=2 \cot \left (2 x \right )\\ q \left (x \right )&=-4 \csc \left (2 x \right )^{2} \end{align*}

Applying change of variables \(\tau = g \left (x \right )\) to (2) gives

\begin{align*} \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+p_{1} \left (\frac {d}{d \tau }y \left (\tau \right )\right )+q_{1} y \left (\tau \right )&=0 \tag {3} \end{align*}

Where \(\tau \) is the new independent variable, and

\begin{align*} p_{1} \left (\tau \right ) &=\frac {\tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {4} \\ q_{1} \left (\tau \right ) &=\frac {q \left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {5} \end{align*}

Let \(p_{1} = 0\). Eq (4) simplifies to

\begin{align*} \tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )&=0 \end{align*}

This ode is solved resulting in

\begin{align*} \tau &= \int {\mathrm e}^{-\int p \left (x \right )d x}d x\\ &= \int {\mathrm e}^{-\int 2 \cot \left (2 x \right )d x}d x\\ &= \int e^{\frac {\ln \left (\csc \left (2 x \right )^{2}\right )}{2}} \,dx\\ &= \int \csc \left (2 x \right )d x\\ &= -\frac {\ln \left (\csc \left (2 x \right )+\cot \left (2 x \right )\right )}{2}\tag {6} \end{align*}

Using (6) to evaluate \(q_{1}\) from (5) gives

\begin{align*} q_{1} \left (\tau \right ) &= \frac {q \left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\\ &= \frac {-4 \csc \left (2 x \right )^{2}}{\csc \left (2 x \right )^{2}}\\ &= -4\tag {7} \end{align*}

Substituting the above in (3) and noting that now \(p_{1} = 0\) results in

\begin{align*} \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+q_{1} y \left (\tau \right )&=0 \\ \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )-4 y \left (\tau \right )&=0 \end{align*}

The above ode is now solved for \(y \left (\tau \right )\).

This is second order with constant coefficients homogeneous ODE. In standard form the ODE is

\[ A y''(\tau ) + B y'(\tau ) + C y(\tau ) = 0 \]

Where in the above \(A=1, B=0, C=-4\). Let the solution be \(y \left (\tau \right )=e^{\lambda \tau }\). Substituting this into the ODE gives

\[ \lambda ^{2} {\mathrm e}^{\tau \lambda }-4 \,{\mathrm e}^{\tau \lambda } = 0 \tag {1} \]

Since exponential function is never zero, then dividing Eq(2) throughout by \(e^{\lambda \tau }\) gives

\[ \lambda ^{2}-4 = 0 \tag {2} \]

Equation (2) is the characteristic equation of the ODE. Its roots determine the general solution form.Using the quadratic formula

\[ \lambda _{1,2} = \frac {-B}{2 A} \pm \frac {1}{2 A} \sqrt {B^2 - 4 A C} \]

Substituting \(A=1, B=0, C=-4\) into the above gives

\begin{align*} \lambda _{1,2} &= \frac {0}{(2) \left (1\right )} \pm \frac {1}{(2) \left (1\right )} \sqrt {0^2 - (4) \left (1\right )\left (-4\right )}\\ &= \pm 2 \end{align*}

Hence

\begin{align*} \lambda _1 &= + 2 \\ \lambda _2 &= - 2 \\ \end{align*}

Which simplifies to

\begin{align*} \lambda _1 &= 2 \\ \lambda _2 &= -2 \\ \end{align*}

Since roots are distinct, then the solution is

\begin{align*} y \left (\tau \right ) &= c_1 e^{\lambda _1 \tau } + c_2 e^{\lambda _2 \tau } \\ y \left (\tau \right ) &= c_1 e^{\left (2\right )\tau } +c_2 e^{\left (-2\right )\tau } \\ \end{align*}

Or

\[ y \left (\tau \right ) =c_1 \,{\mathrm e}^{2 \tau }+c_2 \,{\mathrm e}^{-2 \tau } \]

The above solution is now transformed back to \(y\) using (6) which results in

\[ y = \frac {c_1}{\csc \left (2 x \right )+\cot \left (2 x \right )}+c_2 \left (\csc \left (2 x \right )+\cot \left (2 x \right )\right ) \]

Summary of solutions found

\begin{align*} y &= \frac {c_1}{\csc \left (2 x \right )+\cot \left (2 x \right )}+c_2 \left (\csc \left (2 x \right )+\cot \left (2 x \right )\right ) \\ \end{align*}

✓ Maple. Time used: 0.102 (sec). Leaf size: 17

ode:=diff(diff(y(x),x),x)*sin(2*x)^2+diff(y(x),x)*sin(4*x)-4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \csc \left (2 x \right )+c_2 \cot \left (2 x \right ) \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power \ 
@ Moebius 
-> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x),\ 
 dx)) * 2F1([a1, a2], [b1], f) 
-> Trying changes of variables to rationalize or make the ODE simpler 
   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)] 
   <- linear_1 successful 
   Change of variables used: 
      [x = 1/4*arccos(t)] 
   Linear ODE actually solved: 
      -u(t)+(3*t^2-2*t-1)*diff(u(t),t)+(2*t^3-2*t^2-2*t+2)*diff(diff(u(t),t),t)\ 
 = 0 
<- change of variables successful
 

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 29

ode=D[y[x],{x,2}]*Sin[2*x]^2+D[y[x],x]*Sin[4*x]-4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_1-i c_2 \cos (2 x)}{\sqrt {\sin ^2(2 x)}} \end{align*}

✗ Sympy

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

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