4.27 Problems 1301 to 1350

   4.27.1 \(-2 a \left (1-2 a x^2\right ) y(x)-4 a x y'(x)+y''(x)=0\)
   4.27.2 \(x^2 \left (-y'(x)\right )+y''(x)+x y(x)=0\)
   4.27.3 \(x^2 \left (-y'(x)\right )+y''(x)+x y(x)=x\)
   4.27.4 \(x^2 y'(x)+y''(x)-4 x y(x)=0\)
   4.27.5 \(x^4 y'(x)-x^3 y(x)+y''(x)=0\)
   4.27.6 \(a (k+1) x^{k-1} y(x)+a x^k y'(x)+y''(x)=0\)
   4.27.7 \(a k x^{k-1} y(x)+a x^k y'(x)+y''(x)=0\)
   4.27.8 \(-a x^{k-1} y(x)+a x^k y'(x)+y''(x)=0\)
   4.27.9 \(a x^k y'(x)+b x^{k-1} y(x)+y''(x)=0\)
   4.27.10 \(y''(x)-\cot (x) y'(x)+2 y(x)=0\)
   4.27.11 \(k (k+1) y(x)+y''(x)+\cot (x) y'(x)=0\)
   4.27.12 \(y''(x)+\cot (x) y'(x)-y(x) \csc ^2(x)=0\)
   4.27.13 \(y(x) \left (p (p+1)-k^2 \csc ^2(x)\right )+y''(x)+\cot (x) y'(x)=0\)
   4.27.14 \(y(x) \left (\text {a0}+4 \text {a1} \sin ^2(x)-\text {a2} \csc ^2(x)\right )+y''(x)+\cot (x) y'(x)=0\)
   4.27.15 \(y''(x)+2 \cot (x) y'(x)+3 y(x)=0\)
   4.27.16 \(y''(x)+2 \cot (x) y'(x)+3 y(x)=e^x \csc (x)\)
   4.27.17 \(a \cot (x) y'(x)+y(x) \left (b+k^2 \cos ^2(x)\right )+y''(x)=0\)
   4.27.18 \(y(x) \left (a \cot ^2(x)+b \cot (x) \csc (x)+c \csc ^2(x)\right )+k \cot (x) y'(x)+y''(x)=0\)
   4.27.19 \(y''(x)-\cot (2 x) y'(x)+2 y(x)=0\)
   4.27.20 \(a y(x) \tan ^2(x)+y''(x)-2 \cot (2 x) y'(x)=0\)
   4.27.21 \(a \cot (b x) y'(x)+c y(x)+y''(x)=0\)
   4.27.22 \(\left (b^2-a^2\right ) y(x)+2 a \cot (a x) y'(x)+y''(x)=0\)
   4.27.23 \(a y(x) \tan ^2\left (\frac {x}{2}\right )+y''(x)-\csc (x) y'(x)=0\)
   4.27.24 \(y''(x)+y'(x) (\cot (x)+\csc (x))=a \csc (x)+1\)
   4.27.25 \(y''(x)-\csc (2 x) y'(x)+y(x) \left (\sin ^2(x)+2\right ) \csc ^2(x)=0\)
   4.27.26 \(a y(x) \csc ^2(x)+y''(x)+(\cos (x)+2) \csc (x) y'(x)=0\)
   4.27.27 \(y''(x)-(3 \cos (x)+2) \csc (x) y'(x)-2 y(x) (\cos (x)+1) \sec (x)=0\)
   4.27.28 \(y''(x)-y'(x) (\cot (x)-\sin (x))+y(x) \sin ^2(x)=0\)
   4.27.29 \(y''(x)-\sin (x) y'(x)+y(x) (-\cos (x))=a-x+x \log (x)\)
   4.27.30 \(-2 \csc (2 x) \left (1-a \sin ^2(x)\right ) y'(x)+b y(x) \tan ^2(x)+y''(x)=0\)
   4.27.31 \(a y(x) \cos ^2(x)+y''(x)+\tan (x) y'(x)=0\)
   4.27.32 \(a y(x) \cot ^2(x)+y''(x)+\tan (x) y'(x)=0\)
   4.27.33 \(-a (a+1) y(x) \csc ^2(x)+y''(x)-\tan (x) y'(x)=0\)
   4.27.34 \(y(x) \left (a \cos ^2(x)-\sec ^2(x)\right )+y''(x)-\tan (x) y'(x)=0\)
   4.27.35 \(y''(x)+2 \tan (x) y'(x)-y(x)=0\)
   4.27.36 \(y''(x)+2 \tan (x) y'(x)-y(x)=(x+1) \sec (x)\)
   4.27.37 \(y''(x)+2 \tan (x) y'(x)+3 y(x)=0\)
   4.27.38 \(b y(x)+y''(x)-2 \tan (x) y'(x)=0\)
   4.27.39 \(-\left (a^2+1\right ) y(x)+y''(x)-2 \tan (x) y'(x)=0\)
   4.27.40 \(-\left (a^2+1\right ) y(x)+y''(x)-2 \tan (x) y'(x)=\sin (x)\)
   4.27.41 \(a \tan (x) y'(x)+b y(x)+y''(x)=0\)
   4.27.42 \(y''(x)-\left (2 e^x+1\right ) y'(x)+e^{2 x} y(x)=0\)
   4.27.43 \(y(x) \left (\text {a0}+4 \text {a1} \sinh ^2(x)-\text {a2} \text {csch}^2(x)\right )+y''(x)+\coth (x) y'(x)=0\)
   4.27.44 \(y(x) \left (\text {a0}+4 \text {a1} \cosh ^2(x)-\text {a2} \text {sech}^2(x)\right )+y''(x)+\tanh (x) y'(x)=0\)
   4.27.45 \(b y(x)+y''(x)+2 \tanh (x) y'(x)=0\)
   4.27.46 \(a \tanh (x) y'(x)+b y(x)+y''(x)=0\)
   4.27.47 \(f(x) y'(x)+y''(x)=0\)
   4.27.48 \(a k x^{k-1} y(x)+2 a x^k y'(x)+2 y''(x)=0\)
   4.27.49 \(3 y''(x)-10 y'(x)+3 y(x)=0\)
   4.27.50 \(4 y''(x)=\left (a+x^2\right ) y(x)\)