81.3.11 problem 11

Internal problem ID [18629]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter III. Ordinary differential equations of the first order and first degree. Exercises at page 33
Problem number : 11
Date solved : Tuesday, January 28, 2025 at 12:04:44 PM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 49

dsolve(diff(y(x),x)+cos(x)*y(x)=y(x)^n*sin(2*x),y(x), singsol=all)
 

\[ y \left (x \right ) = \left (\frac {{\mathrm e}^{\sin \left (x \right ) \left (n -1\right )} c_{1} n -{\mathrm e}^{\sin \left (x \right ) \left (n -1\right )} c_{1} +2 n \sin \left (x \right )-2 \sin \left (x \right )+2}{n -1}\right )^{-\frac {1}{n -1}} \]

✓ Solution by Mathematica

Time used: 6.323 (sec). Leaf size: 36

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

\[ y(x)\to \left (c_1 e^{(n-1) \sin (x)}+\frac {2}{n-1}+2 \sin (x)\right ){}^{\frac {1}{1-n}} \]