60.1.207 problem 208

Internal problem ID [10221]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 208
Date solved : Monday, January 27, 2025 at 06:39:46 PM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime } y+a y^{2}-b \cos \left (x +c \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.034 (sec). Leaf size: 106

dsolve(y(x)*diff(y(x),x)+a*y(x)^2-b*cos(x+c)=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {\sqrt {16 \left (a^{2}+\frac {1}{4}\right )^{2} c_{1} {\mathrm e}^{-2 a x}+16 \left (a \cos \left (x +c \right )+\frac {\sin \left (x +c \right )}{2}\right ) \left (a^{2}+\frac {1}{4}\right ) b}}{4 a^{2}+1} \\ y &= -\frac {\sqrt {16 \left (a^{2}+\frac {1}{4}\right )^{2} c_{1} {\mathrm e}^{-2 a x}+16 \left (a \cos \left (x +c \right )+\frac {\sin \left (x +c \right )}{2}\right ) \left (a^{2}+\frac {1}{4}\right ) b}}{4 a^{2}+1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.372 (sec). Leaf size: 84

DSolve[y[x]*D[y[x],x]+a*y[x]^2-b*Cos[x+c]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -e^{-a x} \sqrt {2 \int _1^xb e^{2 a K[1]} \cos (c+K[1])dK[1]+c_1} \\ y(x)\to e^{-a x} \sqrt {2 \int _1^xb e^{2 a K[1]} \cos (c+K[1])dK[1]+c_1} \\ \end{align*}