82.12.27 problem Ex. 29

Internal problem ID [18792]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Examples on chapter II at page 29
Problem number : Ex. 29
Date solved : Tuesday, January 28, 2025 at 12:17:36 PM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.596 (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 (\cos \left (x \right ) b +\frac {\sin \left (x \right )}{2}\right ) \left (b^{2}+\frac {1}{4}\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 (\cos \left (x \right ) b +\frac {\sin \left (x \right )}{2}\right ) \left (b^{2}+\frac {1}{4}\right ) a}}{4 b^{2}+1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 4.991 (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*}