\(a=k x \text { and } \frac{v d v}{d x}=a\)

\(\Rightarrow \int \limits_u^v v d v=\int \limits_0^x a d x=\int \limits_0^x k x d x\)

\(\left.\Rightarrow \frac{v^2}{2}\right|_u ^v=\left.\frac{k x^2}{2}\right|_0 ^x\)

\(\Rightarrow v^2-u^2=k x^2 \Rightarrow v^2=u^2+k x^2\)