solve the initial value problem. dy/dx = 7x^(-10/11), y(-1)= -3 the function y(x) satisfying dy/dx =…

Explanation:

Step1: Integrate the derivative

Integrate $\frac{dy}{dx}=7x^{-\frac{10}{11}}$ with respect to $x$. Using the power - rule for integration $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $y=\int7x^{-\frac{10}{11}}dx=7\times\frac{x^{-\frac{10}{11}+1}}{-\frac{10}{11}+1}+C$. Simplify the exponent and the coefficient: $y = 7\times\frac{x^{\frac{1}{11}}}{\frac{1}{11}}+C=77x^{\frac{1}{11}}+C$.

Step2: Use the initial condition

Substitute $x=-1$ and $y = - 3$ into $y = 77x^{\frac{1}{11}}+C$. We get $-3=77\times(-1)^{\frac{1}{11}}+C$. Since $(-1)^{\frac{1}{11}}=-1$, then $-3=-77 + C$. Solve for $C$: $C=-3 + 77=74$.

Answer:

$y(x)=77x^{\frac{1}{11}}+74$