Consider the differential equation dy/dx = x^4(y - 2). Find the particular solution y = f(x) to the given differential equation with the initial condition f(0) = 0.
Is this y=e^(x^5/5)+4?
To solve the differential equation dy/dx = x^4(y - 2), we will use the method of separation of variables.
Step 1: Rewrite the equation in the form dy/(y - 2) = x^4dx.
Step 2: Integrate both sides of the equation:
∫(dy/(y - 2)) = ∫(x^4dx).
Step 3: Evaluate the integrals:
ln |y - 2| = (1/5)x^5 + C, where C is the constant of integration.
Step 4: Raise both sides as powers of e:
|y - 2| = e^((1/5)x^5 + C).
Step 5: Remove the absolute value by considering two cases:
Case 1: y - 2 > 0, which leads to y - 2 = e^((1/5)x^5 + C).
Case 2: y - 2 < 0, which leads to -(y - 2) = e^((1/5)x^5 + C).
Step 6: Simplify both cases:
Case 1: y = e^((1/5)x^5 + C) + 2.
Case 2: y = -e^((1/5)x^5 + C) + 2.
Step 7: Apply the initial condition f(0) = 0:
Plugging in x = 0 and y = 0 into either case, we get:
Case 1: 0 = e^(C) + 2.
Case 2: 0 = -e^(C) + 2.
Solving the equations for both cases, we find:
Case 1: e^(C) = -2, which is not possible.
Case 2: e^(C) = 2, which gives C = ln(2).
Step 8: Substitute the value of C in either case:
Substituting C = ln(2) into either case, we get:
Case 1: y = e^((1/5)x^5 + ln(2)) + 2.
Case 2: y = -e^((1/5)x^5 + ln(2)) + 2.
So, the particular solution to the differential equation with the initial condition f(0) = 0 is y = e^((1/5)x^5 + ln(2)) + 2.
To find the particular solution to the given differential equation with the initial condition, we can follow these steps:
Step 1: Separate the variables by moving all terms involving y to one side and involving x to the other side.
dy / (y - 2) = x^4 dx
Step 2: Integrate both sides of the equation separately.
∫ (1 / (y - 2)) dy = ∫ x^4 dx
Step 3: Evaluate the integrals.
ln|y - 2| = (1/5) * x^5 + C
where C is the constant of integration.
Step 4: Solve for y.
Taking the exponential of both sides to eliminate the natural logarithm:
|y - 2| = e^((1/5) * x^5 + C)
Now, we consider the initial condition f(0) = 0. Substituting this into the equation:
|0 - 2| = e^((1/5) * 0^5 + C)
2 = e^C
Since the exponential function is always positive, we remove the absolute value:
y - 2 = e^((1/5) * x^5 + C) or 2 - y = e^((1/5) * x^5 + C)
Simplifying, we get two possible solutions:
y = 2 + e^((1/5) * x^5 + C) or y = 2 - e^((1/5) * x^5 + C)
However, without a specific value for C, we cannot determine whether the solution is y = 2 + e^((1/5) * x^5 + C) or y = 2 - e^((1/5) * x^5 + C). Therefore, we cannot confirm if y = e^(x^5/5) + 4 is the correct particular solution without a specific value for C.
dy/(y-2) = x^4 dx
ln(y-2) = x^5/5 + ln(c)
y-2 = c e^(x^5/5)
y(0)=0, so c=-2.
y = 2-2e^(1/5 x^5)
I suspect you did
ln(y-2) = x^5/5 + c
y-2 = e^(x^5/5) + c
That still does not give your answer. Too bad you didn't show your work ...