Find the solution of the differential equation that satisfies the given initial condition.
dp/dt=2 sqrt(pt), P(1)=5
my answer: P=(2/3t^(3/2)+(15sqrt(5)-10)/15)^2
how is it wrong?
To solve the differential equation, we can separate variables and integrate both sides.
Given:
dp/dt = 2√(pt)
P(1) = 5
On separating variables, we have:
dp/√(p) = 2√(t) dt
Integrating both sides:
∫dp/√(p) = 2 ∫√(t) dt
To integrate the left side, we can rewrite it as:
∫p^(-1/2) dp
Using the power rule of integration, integrating the left side yields:
2√(p) = (2/3) t^(3/2) + C
To find the value of the constant, plug in the given initial condition P(1) = 5:
2√(5) = (2/3)(1)^(3/2) + C
2√(5) = 2/3 + C
C = 2√(5) - 2/3
Substituting back into the equation, we get:
2√(p) = (2/3) t^(3/2) + 2√(5) - 2/3
Squaring both sides to solve for p:
4p = (4/9) t^(3) + (8/3)√(5)t^(3/2) + 8 + (4/9)(5) - (8/3)√(5)t^(3/2) + (4/9)(4/9)
4p = (4/9) t^3 + (16/9)(5) + (4/81)
4p = (4/9) t^3 + (80/9) + (4/81)
p = (1/9) t^3 + (20/9) + (1/81)
Therefore, the solution of the differential equation that satisfies the initial condition P(1) = 5 is:
p = (1/9) t^3 + (20/9) + (1/81)
To find the solution of the differential equation, we will follow these steps:
Step 1: Separate the variables. Start with the differential equation:
dp/dt = 2√(pt)
To solve this equation, we need to separate the variables p and t. Divide both sides by √(pt):
(1/√p)dp = 2√t dt
Step 2: Integrate both sides. Integrate both sides of the equation with respect to their corresponding variables:
∫(1/√p)dp = ∫2√t dt
To integrate the left side, use the substitution u = √p and du = (1/2√p)dp:
2∫du = ∫2√t dt
2u = t^(3/2) + C₁
Step 3: Solve for p. Substitute back the value of u:
2√p = t^(3/2) + C₁
√p = (t^(3/2) + C₁)/2
p = [(t^(3/2) + C₁)/2]^2
Simplifying further:
p = (1/4)(t^(3/2) + C₁)²
Step 4: Apply the initial condition. The initial condition given is P(1) = 5. Substitute t = 1 and p = 5 into the equation:
5 = (1/4)(1^(3/2) + C₁)²
5 = (1/4)(1 + C₁)²
Multiply both sides by 4:
20 = (1 + C₁)²
Take the square root of both sides:
√20 = 1 + C₁
√20 - 1 = C₁
Step 5: Write the final solution. Substitute C₁ back into the equation:
p = (1/4)(t^(3/2) + √20 - 1)²
Therefore, the correct solution to the given differential equation with the initial condition is:
p = (1/4)(t^(3/2) + √20 - 1)²
I hope this explanation helps clarify the correct solution.
dp/dt = 2√(pt)
dp/√p = 2√t dt
2√p = 4/3 t^(3/2) + c
p = (2/3 t^(3/2) + c)^2
Now plug in t=1 to find c.