Want to fact check the solution steps used by my textbook in transforming the above differential equation into an Integral Equation.

Here are the steps:

$$\frac{d^2y}{dx^2}+\lambda y=0; \;\;\; y(0)=0, \; y'(1)=0$$
The solution is $y=\sin \frac{(2n-1)\pi x}{2},\;\;\lambda={[\frac{(2n-1)\pi}{2}]}^2 \;\;\; n=1,2,3...$

Now the step I don't understand is the switch to $\frac{d^2y}{dx^2}=f(x)$

Under the boundary condition specified, they now solved further:
$$y=x\int\_1^x f(u)du+\int\_x^0uf(u)du$$
The Integral formulation becomes:
$$y(x)=\lambda\int\_0^1K(x, u)y(u)du$$
Where:

$K(x, u)=x \;\;\;\;\; 0\leq x\leq u \;\;\;\;\; 1\leq u\leq x$

$K(x, u)=u \;\;\;\;\; u\leq x\leq1 \;\;\;\;\; x\leq u\leq0$

Where did they get it wrong. And what's the right step to take?