Part 1) Find the general solution of the differential equation dy/dx + 4xy^6 = 0
Option 1) . y(x) = �(A + 10x^2)^-1/5
Option 2) y(x) = �(A - 12x^2)^-1/6
Option 3) y(x) = �(A + 12x^2)^1/6
Option 4) y(x) = �(A - 10x^2)^-1/5
Option 5) y(x) = �(A + 10x^2)^-1/6
Part 2) Find the particular solution y0 such that y0(0) = 1/2
Option 1) y0(x) = (32 + 12x^2)^1/5
Option 2) y0(x) = (64 - 12x^2)^1/6
Option 3) y0(x) = (64 + 12x^2)^-1/6
Option 4) y0(x) = (32 + 10x^2)^-1/5
Option 5) y0(x) = (32 - 10x^2)^-1/5
Part 3) For the particular solution y0 in (ii), find the value of y0(1)
Option 1) 77^1/6
Option 2) 44^-1/5
Option 3) 76^1/6
Option 4) 43^1/5
Option 5) 42^-1/5
dy/dx + 4xy^6 = 0
dy/dx = -4xy^6
y^-6 dy = -4x dx
-1/5 y^-5 = -2x^2 + C
or
y^5 = 10x^2 + C
Now use y(0) = 1/2 to find C, and
then find y(1)
Part 1)
To find the general solution of the differential equation dy/dx + 4xy^6 = 0, we can separate variables and integrate both sides.
Separating variables:
dy/y^6 = -4xdx
Integrating both sides:
∫(1/y^6)dy = ∫(-4x)dx
Integrating the left side:
∫(1/y^6)dy = ∫y^-6dy = ∫y^-6(dy) = y^-5/(-5) = -1/5y^5 + C1
Integrating the right side:
∫(-4x)dx = -2x^2 + C2
Combining the two sides:
-1/5y^5 + C1 = -2x^2 + C2
Rearranging the terms:
C1 + 2x^2 = 1/5y^5 + C2
Combining the constants:
C1 - C2 = A (where A = 1/5y^5)
Substituting back in:
C1 + 2x^2 = A
Therefore, the general solution of the differential equation dy/dx + 4xy^6 = 0 is:
y(x) = (A + 2x^2)^(-1/5)
where A is a constant.
Comparing this with the provided options, the correct answer is Option 1) y(x) = (A + 10x^2)^(-1/5).
Part 2)
To find the particular solution y0 such that y0(0) = 1/2, we substitute the initial condition into the general solution and solve for the constant A.
Substituting x = 0 and y = 1/2 into the general solution:
1/2 = (A + 10(0)^2)^(-1/5)
1/2 = A^(-1/5)
2 = A^(1/5)
A = 2^5 = 32
Therefore, the particular solution is y0(x) = (32 + 10x^2)^(-1/5)
Comparing this with the provided options, the correct answer is Option 1) y0(x) = (32 + 10x^2)^(-1/5).
Part 3)
To find the value of y0(1), we substitute x = 1 into the particular solution y0(x) = (32 + 10x^2)^(-1/5) and evaluate it.
y0(1) = (32 + 10(1)^2)^(-1/5)
y0(1) = (32 + 10)^(-1/5)
y0(1) = 42^(-1/5)
Therefore, the correct answer is Option 5) 42^(-1/5).
Part 1) To find the general solution of the given differential equation, we can start by separating variables. The equation is dy/dx + 4xy^6 = 0.
Separating variables, we can write the equation as:
1/y^6 dy = -4x dx
Now, we integrate both sides of the equation:
∫1/y^6 dy = ∫-4x dx
The integral of 1/y^6 with respect to y is -(1/5)y^-5 + C1, where C1 is an arbitrary constant.
The integral of -4x with respect to x is -2x^2 + C2, where C2 is another arbitrary constant.
Therefore, our general solution is:
-(1/5)y^-5 + C1 = -2x^2 + C2
Rearranging the equation, we get:
y^-5 = 10x^2 + C
Taking the fifth root of both sides, we have:
y = (10x^2 + C)^-1/5
Comparing this result with the given options, we can conclude that Option 1) y(x) = (A + 10x^2)^-1/5 is the general solution to the differential equation.
Part 2) Now, we need to find the particular solution y0 such that y0(0) = 1/2. This means that when x = 0, y0 = 1/2.
Substituting x = 0 and y0 = 1/2 into the general solution, we can solve for the constant A:
(1/2) = (A + 10(0)^2)^-1/5
(1/2) = (A + 0)^-1/5
(1/2) = A^-1/5
Multiplying both sides by 2 and taking the fifth power, we have:
1 = 32A
Therefore, A = 1/32.
Substituting this value of A back into the general solution, we can determine the particular solution:
y0(x) = (1/32 + 10x^2)^-1/5
Among the given options, we can see that Option 1) y0(x) = (32 + 12x^2)^1/5 is the particular solution.
Part 3) Finally, we are asked to find the value of y0(1) for the particular solution we found in Part 2.
Substituting x = 1 into y0(x), we have:
y0(1) = (32 + 12(1)^2)^1/5
y0(1) = (32 + 12)^1/5
y0(1) = (44)^1/5
Comparing this value with the given options, we can conclude that Option 4) 43^1/5 is the value of y0(1).