4.2 Problems 51 to 100

   4.2.1 \(y'(x)=x f(x) y(x)+f(x)+y(x)^2\)
   4.2.2 \(y'(x)=(-4 y(x)+x+3)^2\)
   4.2.3 \(y'(x)=(9 y(x)+4 x+1)^2\)
   4.2.4 \(y'(x)=3 \left (a+b y(x)^2+b x\right )\)
   4.2.5 \(y'(x)=a+b y(x)^2\)
   4.2.6 \(y'(x)=a x+b y(x)^2\)
   4.2.7 \(y'(x)=a+b x+c y(x)^2\)
   4.2.8 \(y'(x)=a x^{n-1}+b x^{2 n}+c y(x)^2\)
   4.2.9 \(y'(x)=a x^2+b y(x)^2\)
   4.2.10 \(y'(x)=\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2\)
   4.2.11 \(y'(x)=a y(x)+b y(x)^2+f(x)\)
   4.2.12 \(y'(x)=a (x-y(x)) y(x)+1\)
   4.2.13 \(y'(x)=a y(x)^2+f(x)+g(x) y(x)\)
   4.2.14 \(y'(x)=x y(x) (y(x)+3)\)
   4.2.15 \(y'(x)=-x^3+\left (2 x^2+1\right ) y(x)-x y(x)^2-x+1\)
   4.2.16 \(y'(x)=x \left (x^2 y(x)-y(x)^2+2\right )\)
   4.2.17 \(y'(x)=-(1-x) y(x)^2+(1-2 x) y(x)+x\)
   4.2.18 \(y'(x)=a x y(x)^2\)
   4.2.19 \(y'(x)=x^n \left (a+b y(x)^2\right )\)
   4.2.20 \(y'(x)=a x^m+b x^n y(x)^2\)
   4.2.21 \(y'(x)=y(x) (a+b y(x) \cos (k x))\)
   4.2.22 \(y'(x)=\sin (x) \left (2 \sec ^2(x)-y(x)\right )\)
   4.2.23 \(y'(x)+4 \csc (x)=y(x)^2 \sin (x)+y(x) (3-\cot (x))\)
   4.2.24 \(y'(x)=y(x) \sec (x)+(\sin (x)-1)^2\)
   4.2.25 \(y'(x)+\left (1-y(x)^2\right ) \tan (x)=0\)
   4.2.26 \(y'(x)=f(x)+g(x) y(x)+h(x) y(x)^2\)
   4.2.27 \(y'(x)=f(x) \left (a+b y(x)+c y(x)^2\right )\)
   4.2.28 \(y(x)^2 (a x+y(x))+y'(x)\)
   4.2.29 \(y'(x)=y(x)^2 \left (a e^x+y(x)\right )\)
   4.2.30 \(3 a (y(x)+2 x) y(x)^2+y'(x)=0\)
   4.2.31 \(y'(x)=y(x) \left (a+b y(x)^2\right )\)
   4.2.32 \(y'(x)=\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2+\text {a3} y(x)^3\)
   4.2.33 \(y'(x)=x y(x)^3\)
   4.2.34 \(y'(x)+y(x) \left (1-x y(x)^2\right )=0\)
   4.2.35 \(y'(x)=y(x)^2 (a+b x y(x))\)
   4.2.36 \(y(x)^3 \left (a+4 b^2 x+3 b x^2\right )+y'(x)+3 x y(x)^2=0\)
   4.2.37 \(y'(x)=y(x)^2 \left (x^3 y(x)+1\right )\)
   4.2.38 \(2 x y(x) \left (a x y(x)^2+1\right )+y'(x)=0\)
   4.2.39 \(y'(x)=y(x)^2-a x \left (1-x^{n-1}\right ) y(x)^3\)
   4.2.40 \(y'(x)=a y(x)^2+x y(x)^3 \left (b+c x^{n-1}\right )\)
   4.2.41 \(y'(x)+y(x) \left (y(x)^2 \sec (x)+\tan (x)\right )=0\)
   4.2.42 \(y'(x)+y(x)^3 \tan (x) \sec (x)=0\)
   4.2.43 \(y'(x)=\text {f0}(x)+\text {f1}(x) y(x)+\text {f2}(x) y(x)^2+\text {f3}(x) y(x)^3\)
   4.2.44 \(y'(x)=y(x) \left (y(x)^3 \sec (x)+\tan (x)\right )\)
   4.2.45 \(y'(x)=a x^{\frac {n}{1-n}}+b y(x)^n\)
   4.2.46 \(y'(x)=f(x) y(x)+g(x) y(x)^k\)
   4.2.47 \(y'(x)=f(x)+g(x) y(x)+h(x) y(x)^n\)
   4.2.48 \(y'(x)=f(x) y(x)^m+g(x) y(x)^n\)
   4.2.49 \(y'(x)=\sqrt {\left | y(x)\right | }\)
   4.2.50 \(y'(x)=a+\sqrt {\text {A0}+\text {B0} y(x)}+b y(x)\)