Question

Find the particular solution of the differential equation that satisfies the initial condition(s).

Answer 1

step 1

we have

`f^(\prime)^(\prime)\left(x\right)=x^{\left(-(3)/(2)\right)}`

Find out the integral of the second derivative

`f^(\prime)\left(x\right)=\int x^{\left(-(3)/(2)\right)}dx=-(2)/(√(x))+C`

Find out the value of C

we have

f''(4)=5

`\begin{gathered} 5=-(2)/(√(4))+C \\ 5=-1+C \\ C=6 \end{gathered}`

therefore

`f^(\prime)\left(x\right)=-(2)/(√(x))+6`

step 2

Find out the integral of the first derivative

`f\mleft(x\mright)=\int-(2)/(√(x))+6=-4√(x)+6x+C`

Find out the value of C

we have

f(0)=0

`\begin{gathered} 0=-4√(0)+6\left(0\right)+C \\ C=0 \end{gathered}`

therefore

`f\lparen x)=-4√(x)+6x`