83.8.15 problem 16

Internal problem ID [19141]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Misc examples on chapter II at page 25
Problem number : 16
Date solved : Tuesday, January 28, 2025 at 01:05:16 PM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 98

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

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

✓ Solution by Mathematica

Time used: 4.981 (sec). Leaf size: 112

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

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