Chứng minh rằng mỗi hàm số sau đây thỏa m,ãn hệ thức tương ứng đã chỉ ra
a) \(y = {\left( {x + \sqrt {{x^2} + 1} } \right)^3};\left( {1 + {x^2}} \right)y” + xy’ - 9y = 0\)
b) \(y = 2\sin 2x;{y^{\left( {2n} \right)}} = {\left( { - 1} \right)^n}{2^{2n}}y\)
a)
\(\eqalign{
& y’ = 3{\left( {x + \sqrt {{x^2} + 1} } \right)^2}.\left[ {1 + {x \over {\sqrt {{x^2} + 1} }}} \right] \cr
& y” = 6\left( {x + \sqrt {{x^2} + 1} } \right).\left[ {1 + {x \over {\sqrt {{x^2} + 1} }}} \right] \cr&+ 6\left( {x + \sqrt {{x^2} + 1} } \right).\left[ {1 + {x \over {\sqrt {{x^2} + 1} }}} \right].{x \over {\sqrt {{x^2} + 1} }} \cr&+ 3{\left( {x + \sqrt {{x^2} + 1} } \right)^2}.{1 \over {\left( {{x^2} + 1} \right).\sqrt {{x^2} + 1} }} \cr} \)
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Do đó: \(\left( {1 + {x^2}} \right)y” + xy’ - 9y = 0\)
b) Ta có
\(\eqalign{& y’ = 2\cos 2x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y” = - {2^2}\sin 2x \cr& y”’ = - {2^3}\cos 2x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{y^{\left( 4 \right)}} = {2^4}\sin 2x \cr& {y^{\left( 5 \right)}} = {2^5}\cos 2x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{y^{\left( 6 \right)}} = - {2^6}\sin 2x \cr& {y^{\left( 7 \right)}} = - {2^7}\sin 2x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{y^{\left( 8 \right)}} = {2^8}\sin 2x \cr& ... \cr} \)
Bằng phương pháp quy nạp, dễ dàng chứng minh được
\({y^{\left( {2n} \right)}} = {\left( { - 1} \right)^n}{2^{2n}}\sin 2x = {\left( { - 1} \right)^n}{2^{2n}}y\)