Question

Solve the initial value problemf’(x) = 1/x - 2x + x^1/2; f(4) = 2

Answer 1

Case: Initial value problem

An initial value problem is an ordinary differential equation together with an initial condition that specifies the value of the unknown function at a given point in the domain

To solve the initial value problem, we have a value of the derivative of y when x is known.

y'(x)= value

Given:

f(4)=2

`\begin{gathered} f^(\prime)(x)=(1)/(x)-2x+x^{(1)/(2)} \\ Integrating \\ f(x)=ln(x)-(2x^2)/(2)+\frac{2x^{(3)/(2)}}{3}+C \\ f(x)=ln(x)-x^2+\frac{2x^{(3)/(2)}}{3}+C \\ f(4)=2 \\ f(4)=ln(4)-2(4)^2+\frac{2(2)^{(3)/(2)}}{3}+C \\ 2=1.386-32+1.8856+C \\ C=2-1.3863+32-1.8856 \\ C=30.7281 \end{gathered}`

The resulting equation will be:

`f(x)=\ln x-x^2+\frac{2x^{(3)/(2)}}{3}+30.7281`

Final answer: