4.36 Problems 1751 to 1800

   4.36.1 \(-y(x) \left (4 k^2+\left (4 p^2+1\right ) \left (1-x^2\right )\right )+4 \left (1-x^2\right )^2 y''(x)-8 x \left (1-x^2\right ) y'(x)=0\)
   4.36.2 \(-y(x) \left (4 k^2+\left (1-4 p^2\right ) \left (1-x^2\right )\right )+4 \left (1-x^2\right )^2 y''(x)-8 x \left (1-x^2\right ) y'(x)=0\)
   4.36.3 \(-y(x) \left (a (a+1) (1-x)+b^2 x\right )+4 \left (1-x^2\right ) x^2 y''(x)+2 (1-3 x) (1-x) x y'(x)=0\)
   4.36.4 \((a+b x)^4 y''(x)+y(x)=0\)
   4.36.5 \(y''(x) \left (a+b x+c x^2\right )^2+A y(x)=0\)
   4.36.6 \(x^5 y''(x)+x y'(x)-y(x)=0\)
   4.36.7 \(x^5 y''(x)-\left (1-2 x^3\right ) x y'(x)+\left (1-2 x^3\right ) y(x)=0\)
   4.36.8 \(x \left (a^2-x^2\right ) \left (b^2-x^2\right ) y''(x)+\left (\text {a0}+\text {b0} x^4\right ) y'(x)+x^3 y(x) \left (\text {a1}+\text {b1} x^2+\text {c1} x^4\right )=0\)
   4.36.9 \(a y(x)+x^6 y''(x)-x^5 y'(x)=0\)
   4.36.10 \(x^6 y''(x)+3 x^5 y'(x)+y(x)=0\)
   4.36.11 \(x^3 \left (a+3 x^2\right ) y'(x)+x^6 y''(x)+y(x)=0\)
   4.36.12 \((a-x) (b-x) (c-x) y'(x) \left (\text {a1}+\text {b1} x+\text {c1} x^2\right )+(a-x)^2 (b-x)^2 (c-x)^2 y''(x)+y(x) \left (\text {a2}+\text {b2} x+\text {c2} x^2\right )=0\)
   4.36.13 \(4 x^6 y''(x)+\left (1-2 x^2\right ) y(x)+4 \left (2 x^2+1\right ) x^3 y'(x)=0\)
   4.36.14 \(4 x^6 y''(x)+\left (8 x^4+10 x^2+1\right ) y(x)-4 \left (2 x^2+1\right ) x^3 y'(x)=0\)
   4.36.15 \(\left (4 b-a^2\right ) y(x)+4 x^6 y''(x)+12 x^5 y'(x)=0\)
   4.36.16 \(x^{2 a} y''(x)+a x^{2 a-1} y'(x)+(1-a)^2 y(x)=0\)
   4.36.17 \(a^2 x^{a-1} y(x)+x^{a+1} y''(x)+(1-2 a) x^a y'(x)=0\)
   4.36.18 \(x^2 y''(x) \left (\text {a0}+\text {b0} x^k\right )+x y'(x) \left (\text {a1}+\text {b1} x^k\right )+y(x) \left (\text {a2}+\text {b2} x^k\right )=0\)
   4.36.19 \(\left (1-a^2 \cos ^2(x)\right ) y''(x)+a^2 \sin (x) \cos (x) y'(x)+y(x) \left (\text {a0}+\text {a1} \cos ^2(x)\right )=0\)
   4.36.20 \(-y(x) \left (4 k^2-\left (1-p^2\right ) \sinh ^2(x)\right )+4 \sinh ^2(x) y''(x)+4 \sinh (x) \cosh (x) y'(x)=0\)
   4.36.21 \(y''(x)=0\)
   4.36.22 \(y''(x)=a y(x)\)
   4.36.23 \(y''(x)=6 y(x)^2\)
   4.36.24 \(y''(x)=6 y(x)^2+x\)
   4.36.25 \(y''(x)=a+b x+c y(x)^2\)
   4.36.26 \(y''(x)=2 y(x)^3\)
   4.36.27 \(y''(x)=a+b y(x)+2 y(x)^3\)
   4.36.28 \(y''(x)=a+2 y(x)^3+x y(x)\)
   4.36.29 \(y''(x)=f(x)+g(x) y(x)+2 y(x)^3\)
   4.36.30 \(y''(x)=-2 a b x y(x)+a+2 b^2 y(x)^3\)
   4.36.31 \(y''(x)=\text {a0}+\text {a1} x y(x)+\text {a2} y(x)+\text {a3} y(x)^3\)
   4.36.32 \(y''(x)=\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2+\text {a3} y(x)^3\)
   4.36.33 \(a x^r y(x)^s+y''(x)=0\)
   4.36.34 \(a \sin (y(x))+y''(x)=0\)
   4.36.35 \(a e^{y(x)}+y''(x)=0\)
   4.36.36 \(y''(x)=f(y(x))\)
   4.36.37 \(y(x) \left (f'(x)+2 f(x)^2\right )+3 f(x) y'(x)+y''(x)=2 y(x)^3\)
   4.36.38 \(y''(x)+y(x) y'(x)=0\)
   4.36.39 \(y''(x)+y(x) y'(x)=y(x)^3\)
   4.36.40 \(a y(x)+y''(x)+y(x) y'(x)=y(x)^3\)
   4.36.41 \(y''(x)+y(x) y'(x)=12 f'(x)-12 f(x) y(x)+y(x)^3\)
   4.36.42 \(2 a^2 y(x)+(3 a+y(x)) y'(x)+a y(x)^2+y''(x)=y(x)^3\)
   4.36.43 \(y''(x)=y(x) \left (f'(x)-2 f(x)^2\right )+(3 f(x)-y(x)) y'(x)+f(x) y(x)^2+y(x)^3\)
   4.36.44 \(y''(x)=(3 \text {f1}(x)-y(x)) y'(x)+\text {f1}(x) y(x)^2+\text {f2}(x)+\text {f3}(x) y(x)+y(x)^3\)
   4.36.45 \(y''(x)=y'(x) (\text {f0}(x) y(x)+\text {f1}(x))+\text {g0}(x) y(x)^3+\text {g1}(x) y(x)^2+\text {g2}(x) y(x)+\text {g3}(x)\)
   4.36.46 \(y''(x)=y(x) f'(x)+(f(x)-2 y(x)) y'(x)\)
   4.36.47 \(y''(x)=(f(x)-2 y(x)) y'(x)+f(x) y(x)^2+g(x)\)
   4.36.48 \(y''(x)=(\text {f1}(x)-2 y(x)) y'(x)+\text {f2}(x) y(x)^2+\text {f3}(x)\)
   4.36.49 \(y''(x)=(\text {f1}(x)-2 y(x)) y'(x)+\text {f2}(x) y(x)^2+\text {f3}(x) y(x)+\text {f4}(x)\)
   4.36.50 \(y''(x)=a+4 b^2 y(x)+3 b y(x)^2+3 y(x) y'(x)\)