4.31 Problems 1501 to 1550

   4.31.1 \(x^2 y''(x)-(x+2) x y'(x)+(x+2) y(x)=0\)
   4.31.2 \(x^2 y''(x)-(x+2) x y'(x)+(x+2) y(x)=x^3\)
   4.31.3 \(x^2 y''(x)+(2-x) x y'(x)-(3 x+2) y(x)=0\)
   4.31.4 \(x^2 y''(x)+(x+3) x y'(x)-y(x)=0\)
   4.31.5 \(x^2 y''(x)-2 (x+1) x y'(x)+2 (x+1) y(x)=0\)
   4.31.6 \(a x^2 y'(x)+x^2 y''(x)-2 y(x)=0\)
   4.31.7 \(-x (a x+5) y'(x)+(3 a x+5) y(x)+x^2 y''(x)=0\)
   4.31.8 \(x (\text {a1}+\text {b1} x) y'(x)+y(x) \left (\text {a2}+\text {b2} x+\text {c2} x^2\right )+x^2 y''(x)=0\)
   4.31.9 \(x^3 y'(x)+x^2 y''(x)-\left (2-x^2\right ) y(x)=0\)
   4.31.10 \(x^2 y''(x)+\left (1-x^2\right ) x y'(x)-\left (x^2+1\right ) y(x)=0\)
   4.31.11 \(4 x^3 y'(x)+x^2 y''(x)+\left (4 x^4+2 x^2+1\right ) y(x)=0\)
   4.31.12 \(x y'(x) \left (\text {a0}+\text {b0} x^k\right )+y(x) \left (\text {a1}+\text {b1} x^k+\text {c1} x^{2 k}\right )+x^2 y''(x)=0\)
   4.31.13 \(x y'(x) \left (\text {a0}+\text {a1} x^r+\text {a2} x^s\right )+y(x) \left (\text {b0}+\text {b1} x^r+\text {b2} x^{2 r}+\text {b3} x^s+\text {b4} x^{2 s}+\text {b5} x^{r+s}\right )+x^2 y''(x)=0\)
   4.31.14 \(a y(x)+x^2 y''(x)+2 x^2 \cot (x) y'(x)=0\)
   4.31.15 \(-y(x) (a-x \cot (x))+x^2 y''(x)+x (2 x \cot (x)+1) y'(x)=0\)
   4.31.16 \(a y(x)+x^2 y''(x)-2 x^2 \tan (x) y'(x)=0\)
   4.31.17 \(-y(x) (a+x \tan (x))+x^2 y''(x)+x (1-2 x \tan (x)) y'(x)=0\)
   4.31.18 \(y(x) \left (f(x) \left (\text {a1}+\text {b1} x^k-1\right )+\text {a2}+\text {b2} x^k+\text {c2} x^{2 k}+f'(x)+f(x)^2\right )+x y'(x) \left (\text {a1}+\text {b1} x^k+2 f(x)\right )+x^2 y''(x)=0\)
   4.31.19 \(\left (x^2+1\right ) y''(x)-2 y(x)=0\)
   4.31.20 \(a+\left (x^2+1\right ) y''(x)-x y'(x)=0\)
   4.31.21 \(\left (1-x^2\right ) y''(x)+x y'(x)=x\)
   4.31.22 \(\left (x^2+1\right ) y''(x)-x y'(x)+y(x)=0\)
   4.31.23 \(\left (1-x^2\right ) y''(x)+x y'(x)-y(x)=0\)
   4.31.24 \(\left (1-x^2\right ) y''(x)-x y'(x)+y(x)=0\)
   4.31.25 \(\left (1-x^2\right ) y''(x)-x y'(x)-y(x)=0\)
   4.31.26 \(\left (1-x^2\right ) y''(x)+x y'(x)-y(x)=x \left (1-x^2\right )^{3/2}\)
   4.31.27 \(\left (1-x^2\right ) y''(x)+x y'(x)+3 y(x)=0\)
   4.31.28 \(\left (x^2+1\right ) y''(x)+x y'(x)-4 y(x)=0\)
   4.31.29 \(\left (x^2+1\right ) y''(x)+x y'(x)-4 y(x)=0\)
   4.31.30 \(n^2 y(x)+\left (1-x^2\right ) y''(x)-x y'(x)=0\)
   4.31.31 \(a^2 y(x)+\left (x^2+1\right ) y''(x)+x y'(x)=0\)
   4.31.32 \(a^2 y(x)+\left (1-x^2\right ) y''(x)-x y'(x)=0\)
   4.31.33 \(y(x) \left (a+b x^2\right )+\left (1-x^2\right ) y''(x)-x y'(x)=0\)
   4.31.34 \(y(x) \left (\sum _{m=0}^n a(m) x^{2 m}\right )+\left (1-x^2\right ) y''(x)-x y'(x)=0\)
   4.31.35 \(\left (1-x^2\right ) y''(x)-2 x y'(x)=0\)
   4.31.36 \(a+\left (1-x^2\right ) y''(x)-2 x y'(x)=0\)
   4.31.37 \(\left (x^2+1\right ) y''(x)+2 x y'(x)-2 y(x)=0\)
   4.31.38 \(\left (x^2+1\right ) y''(x)-2 x y'(x)+2 y(x)=0\)
   4.31.39 \(\left (1-x^2\right ) y''(x)+2 x y'(x)-2 y(x)=\left (1-x^2\right )^2\)
   4.31.40 \(n (n+1) y(x)+\left (1-x^2\right ) y''(x)-2 x y'(x)=0\)
   4.31.41 \(n (n+1) y(x)+\left (1-x^2\right ) y''(x)-2 x y'(x)=\frac {2 ((-n-1) x P_n(x)+(n+1) P_{n+1}(x))}{x^2-1}\)
   4.31.42 \(-p (p+1) y(x)+\left (x^2+1\right ) y''(x)+2 x y'(x)=0\)
   4.31.43 \(p (p+1) y(x)+\left (1-x^2\right ) y''(x)-2 x y'(x)=0\)
   4.31.44 \(n (n+2) y(x)+\left (1-x^2\right ) y''(x)-3 x y'(x)=0\)
   4.31.45 \(-a y(x)+\left (1-x^2\right ) y''(x)-3 x y'(x)=0\)
   4.31.46 \(\left (x^2+1\right ) y''(x)+4 x y'(x)+2 y(x)=2 (\cos (x)-x)\)
   4.31.47 \(\left (x^2+1\right ) y''(x)-4 x y'(x)+6 y(x)=0\)
   4.31.48 \(\left (1-x^2\right ) y''(x)-\left (x^2+1\right ) y(x)-4 x y'(x)=0\)
   4.31.49 \(\left (1-x^2\right ) y''(x)-6 x y'(x)-4 y(x)=0\)
   4.31.50 \(n y(x) (a+b+n+1)+(-x (a+b+2)-a+b) y'(x)+\left (1-x^2\right ) y''(x)=0\)