What is (d^2)/(dt^2)((v)(av(t)-bv(t)ln(v))) ?

The answer to (d^2)/(dt^2)((v)(av(t)-bv(t)ln(v))) is

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(d^2)/(dt^2)((v)(av(t)-bv(t)ln(v)))

\frac{d ^{2}}{d t ^{2}} ((v (t)) (a v (t) - b v (t) ln (v (t))))

2 a \frac{d}{d t} (v (t))^{2} + 2 a v (t) \frac{d ^{2}}{d t ^{2}} (v (t)) - 3 b \frac{d}{d t} (v (t))^{2} - 2 b ln (v (t)) \frac{d}{d t} (v (t))^{2} - 2 b ln (v (t)) v (t) \frac{d ^{2}}{d t ^{2}} (v (t)) - b v (t) \frac{d ^{2}}{d t ^{2}} (v (t))

Solution steps

\frac{d ^{2}}{d t ^{2}} ((v (t)) (a v (t) - b v (t) ln (v (t))))

\frac{d}{d t} (v (t) (a v (t) - b v (t) ln (v (t)))) = 2 a v (t) \frac{d}{d t} (v (t)) - 2 b ln (v (t)) v (t) \frac{d}{d t} (v (t)) - b v (t) \frac{d}{d t} (v (t))

= \frac{d}{d t} (2 a v (t) \frac{d}{d t} (v (t)) - 2 b ln (v (t)) v (t) \frac{d}{d t} (v (t)) - b v (t) \frac{d}{d t} (v (t)))

\frac{d}{d t} (2 a v (t) \frac{d}{d t} (v (t)) - 2 b ln (v (t)) v (t) \frac{d}{d t} (v (t)) - b v (t) \frac{d}{d t} (v (t))) = 2 a \frac{d}{d t} (v (t))^{2} + 2 a v (t) \frac{d ^{2}}{d t ^{2}} (v (t)) - 3 b \frac{d}{d t} (v (t))^{2} - 2 b ln (v (t)) \frac{d}{d t} (v (t))^{2} - 2 b ln (v (t)) v (t) \frac{d ^{2}}{d t ^{2}} (v (t)) - b v (t) \frac{d ^{2}}{d t ^{2}} (v (t))

= 2 a \frac{d}{d t} (v (t))^{2} + 2 a v (t) \frac{d ^{2}}{d t ^{2}} (v (t)) - 3 b \frac{d}{d t} (v (t))^{2} - 2 b ln (v (t)) \frac{d}{d t} (v (t))^{2} - 2 b ln (v (t)) v (t) \frac{d ^{2}}{d t ^{2}} (v (t)) - b v (t) \frac{d ^{2}}{d t ^{2}} (v (t))

\frac{( 2 a ^{2} - 1 ) ( 4 a ^{4} - 2 a ^{2} + 1 )}{8}

- \frac{- l ( - 2 l ( 3 a - l ^{2} ) - a ) + 3 a}{2}

(y^{2} e^{x y^{2}} + 4 x^{3}) d x + (2 x y e^{x y^{2}} - 3 y^{2}) d y

\frac{x ^{4} + 4 x ^{2} - 9}{4 x ( x ^{2} + 6 x - 9 )}

\frac{( 4 x ^{2} + 5 )}{( x ^{2} + 1 ) ( x ^{2} + 9 )}

What is (d^2)/(dt^2)((v)(av(t)-bv(t)ln(v))) ?
The answer to (d^2)/(dt^2)((v)(av(t)-bv(t)ln(v))) is