4.34 Problems 1651 to 1700

   4.34.1 \(\left (a x^2+1\right ) y''(x)+a x y'(x)+b^2 y(x)=0\)
   4.34.2 \(\left (a x^2+1\right ) y''(x)+b x y'(x)+c y(x)=0\)
   4.34.3 \(\left (1-a^2 x^2\right ) y''(x)-2 a^2 x y'(x)=0\)
   4.34.4 \(\left (1-a^2 x^2\right ) y''(x)-2 a^2 x y'(x)+2 a^2 y(x)=0\)
   4.34.5 \(x (a+b x) y''(x)+2 a y'(x)-2 b y(x)=0\)
   4.34.6 \(y''(x) \left (\text {a0}+\text {b0} x+\text {c0} x^2\right )+(\text {a1}+\text {b1} x) y'(x)+2 \text {a2} y(x)=0\)
   4.34.7 \(\text {a1} (a+b x) y'(x)+(a+b x)^2 y''(x)+\text {a2} y(x)=0\)
   4.34.8 \(x^3 y''(x)=a+b x\)
   4.34.9 \(x^3 y''(x)+x y'(x)-y(x)=0\)
   4.34.10 \(x^3 y''(x)+x y'(x)-2 y(x)=0\)
   4.34.11 \(x^3 y''(x)+2 x y'(x)-y(x)=0\)
   4.34.12 \(\text {a1} x y'(x)+y(x) (\text {a2}+\text {b2} x)+x^3 y''(x)=0\)
   4.34.13 \(y(x) \left (a+b x+c x^2\right )+x^3 y''(x)+x^2 y'(x)=0\)
   4.34.14 \(x^3 y''(x)+3 x^2 y'(x)+x y(x)=0\)
   4.34.15 \(x^3 y''(x)+3 x^2 y'(x)+x y(x)=1\)
   4.34.16 \(a x^2 y'(x)+y(x) (b+c x)+x^3 y''(x)=0\)
   4.34.17 \(\left (\text {a1}+\text {b1} x^2\right ) y'(x)+\text {a2} x y(x)+x^3 y''(x)=0\)
   4.34.18 \(x (\text {a1}+\text {b1} x) y'(x)+\text {a2} y(x)+x^3 y''(x)=0\)
   4.34.19 \(x (\text {a1}+\text {b1} x) y'(x)+y(x) (\text {a2}+\text {b2} x)+x^3 y''(x)=0\)
   4.34.20 \(\left (1-x^3\right ) y''(x)+6 x y(x)=0\)
   4.34.21 \(x \left (1-x^2\right ) y''(x)-y'(x)=0\)
   4.34.22 \(x^3+\left (1-x^2\right ) x y''(x)-y'(x)=0\)
   4.34.23 \(a x^3 y(x)+\left (1-x^2\right ) x y''(x)-y'(x)=0\)
   4.34.24 \(x \left (1-x^2\right ) y''(x)-\left (x^2+7\right ) y'(x)+4 x y(x)=0\)
   4.34.25 \(x \left (x^2+1\right ) y''(x)-2 \left (x^2+1\right ) y'(x)+2 x y(x)=0\)
   4.34.26 \(x \left (x^2+1\right ) y''(x)-2 \left (1-x^2\right ) y'(x)-2 x y(x)=0\)
   4.34.27 \(x \left (1-x^2\right ) y''(x)-2 \left (1-x^2\right ) y'(x)-2 x y(x)=0\)
   4.34.28 \(\left (a+b x^2\right ) y'(x)+c x y(x)+x \left (x^2+1\right ) y''(x)=0\)
   4.34.29 \(\left (a+b x^2\right ) y'(x)+(a-1) x (a+b) y(x)+x \left (1-x^2\right ) y''(x)=0\)
   4.34.30 \(\left (a+b x^2\right ) y'(x)+2 (1-b) x y(x)+x \left (1-x^2\right ) y''(x)=0\)
   4.34.31 \(\left (a-(a+1) x^2\right ) y'(x)+c x y(x)+x \left (1-x^2\right ) y''(x)=0\)
   4.34.32 \(\left (a+b x^2\right ) y'(x)+c x y(x)+x \left (1-x^2\right ) y''(x)=0\)
   4.34.33 \(x \left (x^2+2\right ) y''(x)-y'(x)-6 x y(x)=0\)
   4.34.34 \(x \left (2-x^2\right ) y''(x)-\left (x^2+4 x+2\right ) y(x)-\left (-x^3-3 x^2+2 x+2\right ) y'(x)=0\)
   4.34.35 \(x \left (\text {a0}+x^2\right ) y''(x)+\left (\text {a1}+\text {b1} x^2\right ) y'(x)+\text {a2} x y(x)=0\)
   4.34.36 \(x^2 (x+1) y''(x)+x y'(x)+(x+1)^3 (-y(x))=0\)
   4.34.37 \((1-x) x^2 y''(x)-(x+1) x y'(x)+y(x)=0\)
   4.34.38 \((x+1) x^2 y''(x)-(2 x+1) x y'(x)+(2 x+1) y(x)=0\)
   4.34.39 \((1-x) x^2 y''(x)+2 (2-x) x y'(x)+2 (x+1) y(x)=0\)
   4.34.40 \((1-x) x^2 y''(x)-(4-5 x) x y'(x)+(6-9 x) y(x)=0\)
   4.34.41 \((x+1) x^2 y''(x)+2 (3 x+2) x y'(x)+2 (3 x+1) y(x)=0\)
   4.34.42 \(x (\text {a1}+\text {b1} x) y'(x)+y(x) (\text {a2}+\text {b2} x)+(1-x) x^2 y''(x)=0\)
   4.34.43 \(x^2 (\text {a0}+x) y''(x)+x (\text {a1}+\text {b1} x) y'(x)+y(x) (\text {a2}+\text {b2} x)=0\)
   4.34.44 \((1-x)^2 x y''(x)-2 y(x)=0\)
   4.34.45 \(x \left (x^2+1\right ) y''(x)+x (x+1) y'(x)+y(x)=0\)
   4.34.46 \(x (2-x)^2 y''(x)+2 (2-x) y'(x)+2 y(x)=0\)
   4.34.47 \((1-x) x (a-x) y''(x)+y'(x) \left (x^2 (\text {a0}+\text {a1}+1)+\text {a0} \text {a2}-\text {a0}+\text {a1}+x (\text {a2}+\text {a3})-\text {a3}+1\right )+\text {a0} \text {a1} (x-k) y(x)=0\)
   4.34.48 \((\text {a1}-x) (\text {a2}-x) (\text {a3}-x) y''(x)+y'(x) \left (\text {b0}+\text {b1} x+\text {b2} x^2\right )+y(x) (\text {c0}+\text {c1} x)=0\)
   4.34.49 \(\left (1-2 x^3\right ) y''(x)+6 x^2 y'(x)-6 x y(x)=0\)
   4.34.50 \(2 (1-x) x^2 y''(x)+(3-5 x) x y'(x)-(x+1) y(x)=0\)