Solve the separable differential equation: dy/dt=4y^6
and find the particular solution satisfying the initial condition y(0)=-3
y(t)=?
Integral of dt = Integral of (1/4)y^-6 dy
t = -1/(20 y^-5) + C
0 = 1/20*243 +C
t = -1/(20 y^-5) -1/4860
1/(20 y^-5) = -(1/4860) - t
y^5 = (1/20)/[(-1/4860) - t]
= 243/[-1 -4860 t]
y = -3*[1 +4860 t]^(1/5)
Oh, differential equations, you've done it again! Let's tackle this one step by step, shall we?
dy/dt = 4y^6
To solve a separable differential equation like this, we need to separate the variables. So let's divide both sides of the equation by y^6:
(1/y^6) dy = 4 dt
Now, we integrate both sides to find the antiderivatives:
∫(1/y^6) dy = ∫4 dt
Integrating, we get:
- (1/5) y^-5 = 4t + C
Now, let's solve for y by taking the reciprocal of both sides:
y^-5 = -5 / (4t + C)
To get rid of that pesky negative exponent, we can rewrite the equation as:
y = (-5 / (4t + C))^-1/5
Now, let's use the initial condition y(0) = -3 to find the particular solution. Plugging in t = 0 and y = -3:
-3 = (-5 / (4(0) + C))^-1/5
Simplifying a bit:
-3 = (-5 / C)^-1/5
Now let's raise both sides to the power of -5:
(-3)^-5 = (-5 / C)
Solving for C, we get:
C = 5 / (-3)^-5
We can simplify it further, but hey, why not leave it like that? It adds a little bit of mystique to the equation.
So, the particular solution satisfying the initial condition y(0) = -3 is given by:
y(t) = (-5 / (4t + 5 / (-3)^-5))^-1/5
Voila!
To solve the separable differential equation, we need to separate the variables and integrate both sides.
First, let's separate the variables by moving all the terms involving y to one side and all the terms involving t to the other side:
dy/y^6 = 4dt
Now, we can integrate both sides.
Integrating the left side:
∫ (1/y^6) dy = ∫4 dt
To integrate 1/y^6, we can use the power rule for integration:
∫ 1/y^6 dy = -1/5y^5 + C1
where C1 is the constant of integration.
Integrating the right side:
∫4 dt = 4t + C2
where C2 is the constant of integration.
Now, we set these two results equal to each other:
-1/5y^5 + C1 = 4t + C2
To find the particular solution, we need to use the initial condition y(0) = -3.
Substituting t = 0 and y = -3 into the equation:
-1/5(-3)^5 + C1 = 0 + C2
Simplifying:
243/5 + C1 = C2
Rearranging the equation:
C1 = C2 - 243/5
Now, we substitute C1 back into the general solution:
-1/5y^5 + (C2 - 243/5) = 4t + C2
Simplifying:
-1/5y^5 = 4t + 243/5
Finally, solving for y:
y^5 = -20(4t + 243/5)
y^5 = -80t - 972
Taking the fifth root of both sides:
y = (-80t - 972)^(1/5)
Therefore, the particular solution to the initial value problem y(t) = (-80t - 972)^(1/5) that satisfies the initial condition y(0) = -3 is:
y(t) = (-80t - 972)^(1/5)