# Linear odEs and autonomous systems

#### psalf

##### New member

V' = k1 - k2V2/3, V(0) = V0 > 0, k1 > 0, k2 > 0 ............................................................................................... (1)

(i) Verify that V0(t) = (k1/k2)3/2 is solution to differential equation (1).
(ii) If V(t) is the solution to differential equation (1), show that limt→∞V(t) = V0(t) (geometry proof)
(iii) Prove it's useless to find an explicit formula for the solution to the DE (1) with initial condition V(0) =/= (k1/k2)3/2.
(iv) Verify that
F(x) = arctanh (y3) + ln (y + 1) - ln (y - 1) + ln (y2 + y + 1)/2 - ln (y2 - y + 1)/2 - 3y, y = x1/3
is primitive for f (x) = 1/(1 - x2/3) and analyze the equation F (x) - F (1) = t to solve x in function of t

#### jalexandre

##### New member
What have you tried so far?

#### Subhotosh Khan

##### Super Moderator
Staff member

V' = k1 - k2V2/3, V(0) = V0 > 0, k1 > 0, k2 > 0 ............................................................................................... (1)

(i) Verify that V0(t) = (k1/k2)3/2 is solution to differential equation (1).
(ii) If V(t) is the solution to differential equation (1), show that limt→∞V(t) = V0(t) (geometry proof)
(iii) Prove it's useless to find an explicit formula for the solution to the DE (1) with initial condition V(0) =/= (k1/k2)3/2.
(iv) Verify that
F(x) = arctanh (y3) + ln (y + 1) - ln (y - 1) + ln (y2 + y + 1)/2 - ln (y2 - y + 1)/2 - 3y, y = x1/3
is primitive for f (x) = 1/(1 - x2/3) and analyze the equation F (x) - F (1) = t to solve x in function of t

Please show us what you have tried and exactly where you are stuck.

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