Solve the initial value problem
10(t+1)(dy/dt)+9y= 9t
for t > -1 with y(0)=15
chris or Lisa or whoever --
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To solve the initial value problem
10(t + 1)(dy/dt) + 9y = 9t,
we can use the method of integrating factors. Let's go step by step.
Step 1: Rewrite the equation in the form dy/dt + p(t)y = q(t).
Divide both sides of the equation by 10(t + 1) to get:
dy/dt + (9/10(t + 1))y = (9/10)t.
So, p(t) = (9/10(t + 1)) and q(t) = (9/10)t.
Step 2: Find the integrating factor (μ).
The integrating factor μ is given by μ(t) = e^(∫p(t) dt).
∫(9/10(t + 1)) dt = (9/10) ln|t + 1| + C,
where C is the constant of integration.
Therefore, the integrating factor is μ(t) = e^((9/10) ln|t + 1| + C).
Simplifying further, μ(t) = e^((9/10) ln|t + 1|) * e^C.
Since e^C is a constant, we can write the equation as:
μ(t) = A e^((9/10) ln|t + 1|),
where A = e^C.
Step 3: Multiply the whole equation dy/dt + p(t)y = q(t) by the integrating factor μ(t).
We have:
A e^((9/10) ln|t + 1|) (dy/dt) + A e^((9/10) ln|t + 1|) (9/10(t + 1))y = A e^((9/10) ln|t + 1|) (9/10)t.
The left side becomes d/dt(A e^((9/10) ln|t + 1|) y). By applying the chain rule and simplifying, we get:
d/dt(A e^((9/10) ln|t + 1|) y) = A e^((9/10) ln|t + 1|) (9/10)t.
Step 4: Integrate both sides of the equation.
Integrating both sides, we have:
∫d/dt(A e^((9/10) ln|t + 1|) y) dt = ∫A e^((9/10) ln|t + 1|) (9/10)t dt.
On the left side, we can use the integral and derivative inverses to cancel out the derivatives, resulting in:
A e^((9/10) ln|t + 1|) y = ∫A e^((9/10) ln|t + 1|) (9/10)t dt.
Step 5: Evaluate the integrals.
Integrating the right side, we have:
A e^((9/10) ln|t + 1|) y = A ∫ (9/10)t e^((9/10) ln|t + 1|) dt.
To evaluate this integral, let's substitute u = ln|t + 1|, du = dt/(t + 1):
A e^((9/10) ln|t + 1|) y = A ∫ (9/10) e^(9/10 u) du.
Now integrate:
A e^((9/10) ln|t + 1|) y = A (10/9) e^(9/10 u) + C.
To simplify the equation further, substitute back u = ln|t + 1|:
A e^((9/10) ln|t + 1|) y = A (10/9) e^((9/10) ln|t + 1|) + C.
Step 6: Solve for y.
Now, divide both sides of the equation by A e^((9/10) ln|t + 1|):
y = (10/9) + Ce^(-(9/10) ln|t + 1|).
Step 7: Apply the initial condition y(0) = 15.
Using the initial condition y(0) = 15, we substitute t = 0:
15 = (10/9) + Ce^(-(9/10) ln|0 + 1|).
Since ln|0 + 1| simplifies to ln|1| = 0, the equation becomes:
15 = (10/9) + Ce^0.
Simplifying further, we have:
15 = (10/9) + C.
To solve for C, subtract (10/9) from both sides:
C = 15 - (10/9) = 135/9 - 10/9 = 125/9.
Thus, the equation is:
y = (10/9) + (125/9)e^(-(9/10) ln|t + 1|).
To solve the initial value problem
10(t+1)(dy/dt) + 9y = 9t, for t > -1 with y(0) = 15,
we can use the method of solving linear first-order differential equations, which involves integrating factors. The standard form of a linear first-order differential equation is:
dy/dt + p(t)y = q(t).
In this case, we have:
10(t+1)(dy/dt) + 9y = 9t.
Dividing through by 10(t+1), we get:
(dy/dt) + (9/10(t+1))y = (9t/10(t+1)).
We can rewrite the equation as:
(dy/dt) + (9/(10t+10))y = (9t/(10t+10)).
Now, we need to find the integrating factor, denoted by μ(t), which is given by:
μ(t) = e^(∫ (9/(10t+10)) dt).
To find the integral, we can use the substitution method. Let u = 10t + 10, then du/dt = 10. Rearranging, we have dt = du/10. Substituting back, we get:
μ(t) = e^(∫ (9/u) (du/10)) = e^(∫ (9/10u) du).
Simplifying further, we have:
μ(t) = e^(9/10 ∫ (1/u) du) = e^(9/10 ln|u|) = (u^(9/10)).
Substituting u = 10t + 10, we have:
μ(t) = (10t + 10)^(9/10).
Now, multiplying both sides of the differential equation by the integrating factor, we get:
(10t + 10)^(9/10)(dy/dt) + (9/(10t+10))(10t+10)^(9/10)y = (9t/(10t+10))(10t+10)^(9/10).
Simplifying further, we have:
(10t + 10)^(9/10)(dy/dt) + 9y = 9t(10t + 10)^(-1/10).
The left-hand side is now the derivative of (10t + 10)^(9/10)y with respect to t. Therefore, we can rewrite the equation as:
d((10t + 10)^(9/10)y)/dt = 9t(10t + 10)^(-1/10).
Now, integrating both sides with respect to t, we have:
∫ d((10t + 10)^(9/10)y)/dt dt = ∫ 9t(10t + 10)^(-1/10) dt.
Integrating, we get:
(10t + 10)^(9/10)y = 9/2 (10t + 10)^(9/10) + C,
where C is the constant of integration.
Dividing both sides by (10t + 10)^(9/10), we have:
y = (9/2 (10t + 10)^(9/10))/(10t + 10)^(9/10) + C/(10t + 10)^(9/10).
Simplifying, we get:
y = (9/2) + C/(10t + 10)^(9/10).
Now, we can use the initial condition y(0) = 15 to find the particular solution. Substituting t = 0 and y = 15 into the equation, we can solve for C:
15 = (9/2) + C/(10(0) + 10)^(9/10).
15 = (9/2) + C/(10)^(9/10).
Simplifying, we have:
15 - (9/2) = C/10^(9/10).
Multiplying both sides by 10^(9/10), we get:
(15 - (9/2)) * 10^(9/10) = C.
Now we know the value of C, we can substitute it back into the equation for y:
y = (9/2) + C/(10t + 10)^(9/10).
Finally, the solution to the initial value problem is:
y = (9/2) + [(15 - (9/2)) * 10^(9/10)]/(10t + 10)^(9/10).