The following spectral problem is considered \[-y^{\prime \prime }+q(x)y=\lambda y,\ 0<x<1, \eqno(1)\] \[y(0)\cos \beta =y^{\prime }(0)\sin \beta ,\ 0\leq \beta <\pi , \eqno(2)\] \[y(1)=(c\lambda +d)y^{\prime }(1), \eqno(3)\] 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 \[-y^{\prime \prime }=\lambda y,\ 0<x<1,\] \[y(0)=0,\] \[y(1)=\left( \frac{\lambda }{3}+1\right)y^{\prime }(1).\] For this problem \(\lambda _{0}=\lambda _{1}=0\) is a double eigenvalue.The other eigenvalues \(\lambda _{2}<\lambda _{3}<\ldots\) are the solutions of the equation \(\tan{ \sqrt{\lambda }}=\sqrt{\lambda }\left( \frac{\lambda }{3}+1\right)\). 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 \(\tan{ \sqrt{\lambda }}=\sqrt{\lambda }\left( \frac{\lambda }{3}+1\right)\) 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.