4.1 Problems 1 to 50

   4.1.1 \(y'(x)=a f(x)\)
   4.1.2 \(y'(x)=y(x)+x+\sin (x)\)
   4.1.3 \(y'(x)=x^2+2 y(x)+3 \cosh (x)\)
   4.1.4 \(y'(x)=a+b x+c y(x)\)
   4.1.5 \(y'(x)=a \cos (b x+c)+k y(x)\)
   4.1.6 \(y'(x)=a \sin (b x+c)+k y(x)\)
   4.1.7 \(y'(x)=a+b e^{k x}+c y(x)\)
   4.1.8 \(y'(x)=x \left (x^2-y(x)\right )\)
   4.1.9 \(y'(x)=x \left (a y(x)+e^{-x^2}\right )\)
   4.1.10 \(y'(x)=x^2 \left (a x^3+b y(x)\right )\)
   4.1.11 \(y'(x)=a x^n y(x)\)
   4.1.12 \(y'(x)=y(x) \cos (x)+\sin (x) \cos (x)\)
   4.1.13 \(y'(x)=y(x) \cos (x)+e^{\sin (x)}\)
   4.1.14 \(y'(x)=y(x) \cot (x)\)
   4.1.15 \(y'(x)=1-y(x) \cot (x)\)
   4.1.16 \(y'(x)=x \csc (x)-y(x) \cot (x)\)
   4.1.17 \(y'(x)=y(x) (\cot (x)+2 \csc (2 x))\)
   4.1.18 \(y'(x)=\sec (x)-y(x) \cot (x)\)
   4.1.19 \(y'(x)=y(x) \cot (x)+e^x \sin (x)\)
   4.1.20 \(y'(x)+2 y(x) \cot (x)+\csc (x)=0\)
   4.1.21 \(y'(x)=4 x \csc (x) \sec ^2(x)-2 y(x) \cot (2 x)\)
   4.1.22 \(y'(x)=2 \left (\cos (2 x) \cot ^2(x)-y(x) \csc (2 x)\right )\)
   4.1.23 \(y'(x)=4 x \csc (x) \left (y(x)+\sin ^3(x)\right )\)
   4.1.24 \(y'(x)=4 x \csc (x) \left (y(x)-\tan ^2(x)+1\right )\)
   4.1.25 \(y'(x)=y(x) \sec (x)\)
   4.1.26 \(y'(x)+\tan (x)=(1-y(x)) \sec (x)\)
   4.1.27 \(y'(x)=y(x) \tan (x)\)
   4.1.28 \(y'(x)=y(x) \tan (x)+\cos (x)\)
   4.1.29 \(y'(x)=\cos (x)-y(x) \tan (x)\)
   4.1.30 \(y'(x)=\sec (x)-y(x) \tan (x)\)
   4.1.31 \(y'(x)=y(x) \tan (x)+\sin (2 x)\)
   4.1.32 \(y'(x)=\sin (2 x)-y(x) \tan (x)\)
   4.1.33 \(y'(x)=2 y(x) \tan (x)+\sin (x)\)
   4.1.34 \(y'(x)=2 (y(x) \tan (2 x)+\sec (2 x)+1)\)
   4.1.35 \(y'(x)=3 y(x) \tan (x)+\csc (x)\)
   4.1.36 \(y'(x)=y(x) (a+\sin (\log (x))+\cos (\log (x)))\)
   4.1.37 \(y'(x)=6 e^{2 x}-y(x) \tanh (x)\)
   4.1.38 \(y'(x)=y(x) f'(x)+f(x) f'(x)\)
   4.1.39 \(y'(x)=f(x)+g(x) y(x)\)
   4.1.40 \(y'(x)=x^2-y(x)^2\)
   4.1.41 \(f(x)^2+y'(x)=f'(x)+y(x)^2\)
   4.1.42 \(y'(x)-x+1=y(x) (y(x)+x)\)
   4.1.43 \(y'(x)=(y(x)+x)^2\)
   4.1.44 \(y'(x)=(x-y(x))^2\)
   4.1.45 \(y'(x)=(x-y(x))^2+3 (y(x)-x+1)\)
   4.1.46 \(y'(x)=-\left (x^2+1\right ) y(x)+y(x)^2+2 x\)
   4.1.47 \(y'(x)=x \left (x^3+2\right )-\left (2 x^2-y(x)\right ) y(x)\)
   4.1.48 \(y'(x)=x \left (2-x^3\right )+\left (2 x^2-y(x)\right ) y(x)+1\)
   4.1.49 \(y'(x)=\cos (x)-y(x) (\sin (x)-y(x))\)
   4.1.50 \(y'(x)=y(x) (y(x)+\sin (2 x))+\cos (2 x)\)