# `f''(x) = 2, f'(2) = 5, f(2) = 10` Find the particular solution that satisfies the differential equation.

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Expert Answers

gsarora17 | Certified Educator

`f''(x)=2`

`f'(x)=int2dx`

`f'(x)=2x+C_1`

Now let's find constant C_1 , given f'(2)=5

`5=2(2)+C_1`

`5=4+C_1`

`C_1=1`

`:.f'(x)=2x+1`

`f(x)=int(2x+1)dx`

`f(x)=(2x^2)/2+x+C_2`

`f(x)=x^2+x+C_2`

Now let's find constant C_2 , given f(2)=10

`10=2^2+2+C_2`

`10=4+2+C_2`

`C_2=10-6=4`

`:.f(x)=x^2+x+4`