74.4.21 problem 21

Internal problem ID [15914]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number : 21
Date solved : Tuesday, January 28, 2025 at 08:20:26 AM
CAS classification : [_separable]

\begin{align*} \left (2-\frac {5}{y^{2}}\right ) y^{\prime }+4 \cos \left (x \right )^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.058 (sec). Leaf size: 107

dsolve((2-5/y(x)^2)*diff(y(x),x)+4*cos(x)^2=0,y(x), singsol=all)
 

\begin{align*} y &= -c_{1} -\frac {x}{2}-\frac {\sin \left (2 x \right )}{4}-\frac {\sqrt {-158+16 \left (x +2 c_{1} \right ) \sin \left (2 x \right )+16 x^{2}+64 c_{1} x +64 c_{1}^{2}-2 \cos \left (4 x \right )}}{8} \\ y &= -c_{1} -\frac {x}{2}-\frac {\sin \left (2 x \right )}{4}+\frac {\sqrt {-158+16 \left (x +2 c_{1} \right ) \sin \left (2 x \right )+16 x^{2}+64 c_{1} x +64 c_{1}^{2}-2 \cos \left (4 x \right )}}{8} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.370 (sec). Leaf size: 200

DSolve[(2-5/y[x]^2)*D[y[x],x]+4*Cos[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{4} \left (\int _1^x-4 \cos ^2(K[1])dK[1]-\sqrt {-40+\left (\int _1^x-4 \cos ^2(K[1])dK[1]+c_1\right ){}^2}+c_1\right ) \\ y(x)\to \frac {1}{4} \left (\int _1^x-4 \cos ^2(K[1])dK[1]+\sqrt {-40+\left (\int _1^x-4 \cos ^2(K[1])dK[1]+c_1\right ){}^2}+c_1\right ) \\ y(x)\to 0 \\ y(x)\to \frac {1}{4} \left (\int _1^x-4 \cos ^2(K[1])dK[1]-\sqrt {\int _1^x-4 \cos ^2(K[1])dK[1]{}^2-40}\right ) \\ y(x)\to \frac {1}{4} \left (\int _1^x-4 \cos ^2(K[1])dK[1]+\sqrt {\int _1^x-4 \cos ^2(K[1])dK[1]{}^2-40}\right ) \\ \end{align*}