Question

Find an equation of the curve that satisfies dy/dx = 102yx^16 and whose y-intercept is 6. y(x) =

Answer 1

To find the equation of the curve satisfying the given differential equation with a y-intercept of 6, we separated variables, integrated both sides, and used the initial condition to find the constant of integration. The final equation is
`y(x) = e^((102)/(17)x^(17) + ln(6)).`

Step-by-step explanation:

The student is asked to find an equation of the curve that satisfies the differential equation
`dy/dx = 102yx^(16)` and has a y-intercept of 6. To solve this, we perform separation of variables, integrating both sides appropriately. First, we separate variables by dividing both sides by y and multiplying both sides by dx:

`\((1)/(y) dy = 102x^(16) dx\)`

Next, we integrate both sides:

`\(\int (1)/(y) dy = \int 102x^(16) dx\)`

`\(\ln |y| = (102)/(17)x^(17) + C\)`

We then exponentiate both sides to solve for y:

`\(y = e^{(102)/(17)x^(17) + C}\)`

Considering the y-intercept of 6, we have y(0) = 6, which gives us C after we plug in x=0:

`\(6 = e^(C)\)`

`\(C = \ln(6)\)`

Therefore, the equation of the curve is:

`\(y(x) = e^{(102)/(17)x^(17) + \ln(6)}\)`

Answer 2

The solution of the given differential equation is:

`y(x)=6e^{6x^(17)}`

Step-by-step explanation:

The given differential equation is in variable separable form and is solved as shown under

`(dy)/(dx)=102* x^(16)* y\\\\(dy)/(y)=102x^(16)dx`

Integrating bot sides we get

`\int (dy)/(y)=\int 102x^(16)dx\\ln(y)=102* (x^(17))/(17)+c\\\\ln(y)=6x^(17)+c\\\\y=e^{6x^(17)+c}\\\\y=e^(c)\cdot e^{6x^(17)}\\\\\therefore y=ke^{6x^(17)}`

where 'k' is a constant whose value can be found by the given condition that

y(0) = 6

`y(0)=ke^{6* 0^(17)}\\\\\therefore k=6`

The final solution is

`y(x)=6e^{6x^(17)}`