4.9 Problems 401 to 450

   4.9.1 \(\sqrt {X} y'(x)=\sqrt {Y}\)
   4.9.2 \(x^{3/2} y'(x)=a+b x^{3/2} y(x)^2\)
   4.9.3 \(\sqrt {x^3+1} y'(x)=\sqrt {y(x)^3+1}\)
   4.9.4 \(\sqrt {(1-x) x (1-a x)} y'(x)=\sqrt {(1-y(x)) y(x) (1-a y(x))}\)
   4.9.5 \(\sqrt {1-x^4} y'(x)=\sqrt {1-y(x)^4}\)
   4.9.6 \(\sqrt {x^4+x^2+1} y'(x)=\sqrt {y(x)^4+y(x)^2+1}\)
   4.9.7 \(\sqrt {X} y'(x)=0\)
   4.9.8 \(\sqrt {X} y'(x)+\sqrt {Y}=0\)
   4.9.9 \(\sqrt {X} y'(x)=\sqrt {Y}\)
   4.9.10 \(\left (x^3+1\right )^{2/3} y'(x)+\left (y(x)^3+1\right )^{2/3}=0\)
   4.9.11 \(\left (\text {a0}+\text {a1} x+4 x^3\right )^{2/3} y'(x)+\left (\text {a0}+\text {a1} y(x)+4 y(x)^3\right )^{2/3}=0\)
   4.9.12 \(X^{2/3} y'(x)=Y^{2/3}\)
   4.9.13 \(\left (a+\cos ^2\left (\frac {x}{2}\right )\right ) y'(x)=y(x) \tan \left (\frac {x}{2}\right ) \left (a-y(x)+\cos ^2\left (\frac {x}{2}\right )+1\right )\)
   4.9.14 \(\left (1-4 \cos ^2(x)\right ) y'(x)=y(x) \left (4 \cos ^2(x)+1\right ) \tan (x)\)
   4.9.15 \((1-\sin (x)) y'(x)+y(x) \cos (x)=0\)
   4.9.16 \(y'(x) (\cos (x)-\sin (x))+y(x) (\sin (x)+\cos (x))=0\)
   4.9.17 \(y'(x) \left (\text {a0}+\text {a1} \sin ^2(x)\right )+\text {a2} x \left (\text {a1} \sin ^2(x)+\text {a3}\right )+\text {a1} y(x) \sin (2 x)=0\)
   4.9.18 \(\left (x-e^x\right ) y'(x)+\left (1-e^x\right ) y(x)+e^x x=0\)
   4.9.19 \(x \log (x) y'(x)=a x (\log (x)+1)-y(x)\)
   4.9.20 \(y(x) y'(x)+x=0\)
   4.9.21 \(e^{x^2} x+y(x) y'(x)=0\)
   4.9.22 \(x^3+y(x) y'(x)+y(x)=0\)
   4.9.23 \(a x+b y(x)+y(x) y'(x)=0\)
   4.9.24 \(y(x) y'(x)+e^{-x} x (y(x)+1)=0\)
   4.9.25 \(f(x)+y(x) y'(x)=g(x) y(x)\)
   4.9.26 \(y(x) y'(x)+y(x)^2+4 x (x+1)=0\)
   4.9.27 \(y(x) y'(x)=a x+b y(x)^2\)
   4.9.28 \(y(x) y'(x)=a y(x)^2+b \cos (c+x)\)
   4.9.29 \(y(x) y'(x)=\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2\)
   4.9.30 \(y(x) y'(x)=a x+b x y(x)^2\)
   4.9.31 \(y(x) y'(x)=\csc ^2(x)-y(x)^2 \cot (x)\)
   4.9.32 \(y(x) y'(x)=\sqrt {a^2+y(x)^2}\)
   4.9.33 \(y(x) y'(x)=\sqrt {y(x)^2-a^2}\)
   4.9.34 \(g(x) f\left (x^2+y(x)^2\right )+y(x) y'(x)+x=0\)
   4.9.35 \((y(x)+1) y'(x)=y(x)+x\)
   4.9.36 \((y(x)+1) y'(x)=x^2 (1-y(x))\)
   4.9.37 \((y(x)+x) y'(x)+y(x)=0\)
   4.9.38 \((x-y(x)) y'(x)=y(x)\)
   4.9.39 \((y(x)+x) y'(x)-y(x)+x=0\)
   4.9.40 \((y(x)+x) y'(x)=x-y(x)\)
   4.9.41 \(1-y'(x)=y(x)+x\)
   4.9.42 \((x-y(x)) y'(x)=y(x) (2 x y(x)+1)\)
   4.9.43 \((y(x)+x) y'(x)+\tan (y(x))=0\)
   4.9.44 \((x-y(x)) y'(x)=\left (e^{-\frac {x}{y(x)}}+1\right ) y(x)\)
   4.9.45 \((y(x)+x+1) y'(x)+3 y(x)+4 x+1=0\)
   4.9.46 \((y(x)+x+2) y'(x)=-y(x)-x+1\)
   4.9.47 \((-y(x)-x+3) y'(x)=-3 y(x)+x+1\)
   4.9.48 \((y(x)-x+3) y'(x)=3 y(x)-4 x+11\)
   4.9.49 \((y(x)+2 x) y'(x)-2 y(x)+x=0\)
   4.9.50 \((-y(x)+2 x+2) y'(x)+3 (-y(x)+2 x+1)=0\)