83.35.9 problem 9

Internal problem ID [19358]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Misc. Exercise on chapter VII. Page 118
Problem number : 9
Date solved : Monday, March 31, 2025 at 07:09:44 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} \sin \left (y\right )^{3} y^{\prime \prime }&=\cos \left (y\right ) \end{align*}

✓ Maple. Time used: 1.161 (sec). Leaf size: 3626

ode:=sin(y(x))^3*diff(diff(y(x),x),x) = cos(y(x)); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.414 (sec). Leaf size: 86

ode=Sin[y[x]]^3*D[y[x],{x,2}]==Cos[y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {\csc ^2(y(x)) (c_1 \cos (2 y(x))+2-c_1) \log ^2\left (\sqrt {2} \sqrt {c_1} \cos (y(x))+\sqrt {c_1 \cos (2 y(x))+2-c_1}\right )}{2 c_1 \left (-\csc ^2(y(x))+c_1\right )}=(x+c_2){}^2,y(x)\right ] \]

✓ Sympy. Time used: 142.689 (sec). Leaf size: 37

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

\[ \left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {C_{1} - \frac {1}{\sin ^{2}{\left (u \right )}}}}\, du = C_{2} + x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {C_{1} - \frac {1}{\sin ^{2}{\left (u \right )}}}}\, du = C_{2} - x\right ] \]