29.15.17 problem 425

Internal problem ID [5023]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 15
Problem number : 425
Date solved : Monday, January 27, 2025 at 10:04:01 AM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 106

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

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

✓ Solution by Mathematica

Time used: 5.731 (sec). Leaf size: 106

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

\begin{align*} y(x)\to -\frac {\sqrt {\left (4 a^2+1\right ) c_1 e^{2 a x}-4 a b \cos (c+x)+2 b \sin (c+x)}}{\sqrt {4 a^2+1}} \\ y(x)\to \frac {\sqrt {\left (4 a^2+1\right ) c_1 e^{2 a x}-4 a b \cos (c+x)+2 b \sin (c+x)}}{\sqrt {4 a^2+1}} \\ \end{align*}