Question

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

Answer 1

Answer:

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

$\frac{dy}{dx}=102\times x^{16}\times y\\\\\frac{dy}{y}=102x^{16}dx$

Integrating bot sides we get

$\int \frac{dy}{y}=\int 102x^{16}dx\\ln(y)=102\times \frac{x^{17}}{17}+c\\\\ln(y)=6x^{17}+c\\\\y=e^{6x^{17}+c}\\\\y=e^{c}\cdot e^{6x^{17}}\\\\\therefore y=ke^{6x^{17}}$

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

y(0) = 6

$y(0)=ke^{6\times 0^{17}}\\\\\therefore k=6$

The final solution is

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