The following spectral problem is considered $$-y^{\prime \prime }+q(x)y=\lambda y,0<x<1,\text{eqno}(1)$$ $$y(0)\cos \beta =y^{\prime }(0)\sin \beta ,0\le \beta <\pi ,\text{eqno}(2)$$ $$y(1)=(c\lambda +d)y^{\prime }(1),\text{eqno}(3)$$ where $c,d$ are real constants and $c>0$, $\lambda$ is the spectral parameter, $q(x)$ is a real valued and continuous function over the interval $[0,1]$.

In the current study we are mainly interested in numerical evaluation of the eigenvalues and the eigenfunctions of special eigenvalue problems such as $$-y^{\prime \prime }=\lambda y,0<x<1,$$ $$y(0)=0,$$ $$y(1)=(\frac{\lambda }{3}+1)y^{\prime }(1).$$ For this problem ${\lambda }_0={\lambda }_1=0$ is a double eigenvalue.The other eigenvalues ${\lambda }_2<{\lambda }_3<\dots$ are the solutions of the equation $\tan \sqrt{\lambda }=\sqrt{\lambda }(\frac{\lambda }{3}+1)$. Eigenfunctions are $y_0=x$, $y_n=\sin \sqrt{{\lambda }_n}x$ $(n\ge 2)$ and an associated function corresponding to $y_0$ is $y_1=-\frac{1}{6}{x}^3+Cx$, where $C$ is an arbitrary constant. The transcendental equation $\tan \sqrt{\lambda }=\sqrt{\lambda }(\frac{\lambda }{3}+1)$ is approximately solved to find approximate values of ${\lambda }_n$ which then used to find formula for $y_n=\sin \sqrt{{\lambda }_n}x$.

We also discuss an example for which the eigenvalue is triple.