Differential Equations
Number of solutions of x + f(x) = 0
MJAT_TS8_P2
Grade 12
Question:
Let $f(x)$ be differentiable with $f'(x)=f(x)+\displaystyle\int_0^2 f(x)\,dx$ and $f(0)=4-3e^2$. Find the number of solutions of $x+f(x)=0$:
Step-by-Step Solution
Key Concept: Let $k=\int_0^2 f(x)dx$. Then $f'=f+k\Rightarrow f(x)=Ce^x-k$. $\int_0^2(Ce^x-k)dx=k\Rightarrow C(e^2-1)-2k=k\Rightarrow C=3k/(e^2-1)$.
$\mathbf{1}$ solution.
Correct Answer: 1