Question

Solve the differential.f"(x)=3xGiven f'(0)=3 and f(1)=4

Answer 1

Given:

`f^{^(\prime)^(\prime)}(x)\text{ =3x}`

Solving the differential equation:

`\begin{gathered} \int f^(\prime)^(\prime)(x)\text{ dx= }\int3x\text{ dx} \\ f^(\prime)(x)\text{ = 3}(x^2)/(2)+c_1 \end{gathered}`

Applying the initial value f'(0) = 3

`f^(\prime)(x)\text{ = }(3)/(2)x^2\text{ + 3}`

Integrating further:

`\begin{gathered} \int f^(\prime)(x)dx\text{ = }\int(3)/(2)x^2\text{ dx + 3}\int dx \\ f(x)\text{ = }(3)/(2)(x^3)/(3)\text{ + 3x + c} \\ f(x)\text{ = }(1)/(2)x^3\text{ + 3x + c} \end{gathered}`

Applying the initial value f(1) =4

`\begin{gathered} 4\text{ = }(1)/(2)(1)^3\text{ + 3\lparen1\rparen + c} \\ 4\text{ = }(1)/(2)\text{ + 3 + c} \\ Solving\text{ for c} \\ c\text{ = 4-}(1)/(2)\text{ -3} \\ c\text{ = }(8-1-6)/(2) \\ c=\text{ }(1)/(2) \end{gathered}`

Hence, the solution is:

`f(x)\text{ = }(1)/(2)x^3\text{ + 3x + }(1)/(2)`