4.8 Problems 351 to 400

   4.8.1 \(x^3 y'(x)=(x+1) y(x)^2\)
   4.8.2 \(x^3 y'(x)+x^2 y(x) \left (1-x^2 y(x)\right )+20=0\)
   4.8.3 \(x^6 \left (-y(x)^2\right )+x^3 y'(x)+(3-2 x) x^2 y(x)+3=0\)
   4.8.4 \(x^3 y'(x)=y(x) \left (2 x^2+y(x)^2\right )\)
   4.8.5 \(x^3 y'(x)=\cos (y(x)) \left (\cos (y(x))-2 x^2 \sin (y(x))\right )\)
   4.8.6 \(x \left (x^2+1\right ) y'(x)=a x^2+y(x)\)
   4.8.7 \(x \left (1-x^2\right ) y'(x)=a x^2+y(x)\)
   4.8.8 \(x \left (x^2+1\right ) y'(x)=a x^3+y(x)\)
   4.8.9 \(x \left (x^2+1\right ) y'(x)=a-x^2 y(x)\)
   4.8.10 \(x \left (x^2+1\right ) y'(x)=\left (1-x^2\right ) y(x)\)
   4.8.11 \(x \left (1-x^2\right ) y'(x)=\left (x^2-x+1\right ) y(x)\)
   4.8.12 \(x \left (1-x^2\right ) y'(x)=a x^3+\left (1-2 x^2\right ) y(x)\)
   4.8.13 \(x \left (1-x^2\right ) y'(x)=\left (1-2 x^2\right ) y(x)+\left (1-x^2\right ) x^3\)
   4.8.14 \(x \left (x^2+1\right ) y'(x)=2 \left (1-2 x^2 y(x)\right )\)
   4.8.15 \(x \left (x^2+1\right ) y'(x)=x-\left (5 x^2+3\right ) y(x)\)
   4.8.16 \(\left (1-x^2\right ) x y'(x)+\left (1-x^2\right ) y(x)^2+x^2=0\)
   4.8.17 \((1-x) x^2 y'(x)=(2-x) x y(x)-y(x)^2\)
   4.8.18 \(2 x^3 y'(x)=y(x) \left (x^2-y(x)^2\right )\)
   4.8.19 \(2 x^3 y'(x)=y(x) \left (a y(x)^2+3 x^2\right )\)
   4.8.20 \(6 x^3 y'(x)=4 x^2 y(x)+(1-3 x) y(x)^4\)
   4.8.21 \(x y'(x) \left (a+b x+c x^2\right )-y(x) \left (a+b x+c x^2\right )+x^2=y(x)^2\)
   4.8.22 \(x^4 y'(x)=y(x) \left (x^3+y(x)\right )\)
   4.8.23 \(a^2+x^4 y'(x)+x^4 y(x)^2=0\)
   4.8.24 \(x^4 y'(x)+x^3 y(x)+\csc (x y(x))=0\)
   4.8.25 \(\left (1-x^4\right ) y'(x)=2 x \left (1-y(x)^2\right )\)
   4.8.26 \(x \left (1-x^3\right ) y'(x)=2 x-\left (1-4 x^3\right ) y(x)\)
   4.8.27 \(x \left (1-x^3\right ) y'(x)=x^2+y(x) (1-2 x y(x))\)
   4.8.28 \(x^2 \left (1-x^2\right ) y'(x)=y(x) \left (x-3 x^3 y(x)\right )\)
   4.8.29 \(x \left (1-2 x^3\right ) y'(x)=2 \left (1-x^3\right ) y(x)\)
   4.8.30 \(\left (y'(x)+y(x)^2\right ) \left (a+b x+c x^2\right )^2+A=0\)
   4.8.31 \(x^5 y'(x)=1-3 x^4 y(x)\)
   4.8.32 \(x \left (1-x^4\right ) y'(x)=\left (1-x^4\right ) y(x)+2 x \left (x^2-y(x)^2\right )\)
   4.8.33 \(x^7 y'(x)+5 x^3 y(x)^2+2 \left (x^2+1\right ) y(x)^3=0\)
   4.8.34 \(x^n y'(x)=a+b x^{n-1} y(x)\)
   4.8.35 \(x^n y'(x)=x^{2 n-1}-y(x)^2\)
   4.8.36 \(y(x) \left (x^{n-1}+y(x)\right )+x^n y'(x)+x^2=0\)
   4.8.37 \((1-n) x^{n-1}+x^{2 n-2}+x^n y'(x)+y(x)^2=0\)
   4.8.38 \(x^n y'(x)=a^2 x^{2 n-2}+b^2 y(x)^2\)
   4.8.39 \(x^n y'(x)=x^{n-1} \left (a x^{2 n}-b y(x)^2+n y(x)\right )\)
   4.8.40 \(x^{2 n} y'(x)=-n x^{n-1}+x^n y(x) \left (x^{2 n} y(x)^2-3 x^n y(x)+1\right )+1\)
   4.8.41 \(x^k y'(x)=a x^m+b y(x)^n\)
   4.8.42 \(\sqrt {x^2+1} y'(x)=2 x-y(x)\)
   4.8.43 \(\sqrt {1-x^2} y'(x)=y(x)^2+1\)
   4.8.44 \(\left (x-\sqrt {x^2+1}\right ) y'(x)=y(x)+\sqrt {y(x)^2+1}\)
   4.8.45 \(\sqrt {a^2+x^2} y'(x)+y(x)+x=\sqrt {a^2+x^2}\)
   4.8.46 \(\sqrt {b^2+x^2} y'(x)=\sqrt {a^2+y(x)^2}\)
   4.8.47 \(\sqrt {b^2-x^2} y'(x)=\sqrt {a^2-y(x)^2}\)
   4.8.48 \(x \sqrt {a^2+x^2} y'(x)=y(x) \sqrt {b^2+y(x)^2}\)
   4.8.49 \(x \sqrt {x^2-a^2} y'(x)=y(x) \sqrt {y(x)^2-b^2}\)
   4.8.50 \(\sqrt {X} y'(x)+\sqrt {Y}=0\)