(Solved): 10. The values of \( \lambda \) such that \( y=e^{\lambda x} \) is a solution of \( y^{\prime \prim ...

10. The values of \( \lambda \) such that \( y=e^{\lambda x} \) is a solution of \( y^{\prime \prime}-4 y^{\prime}+20 y=0 \) are: (a) \( \lambda=-5,4 \) (b) \( \lambda=10,2 \) (c) \( \lambda=5,-4 \) (d) \( \lambda=10,-2 \) (e) None of the above. 11. The values of \( \lambda \) such that \( y=x^{\lambda}+2 x^{3} \) is a solution of \( x^{2} y^{\prime \prime}-2 y=8 x^{3} \) are: (a) \( \lambda=2,-2 \) (b) \( \lambda=1,-2 \) (c) \( \lambda=3 \) (d) \( \lambda=2,-1 \) (e) None of the above. 12. The values of \( \lambda \) such that \( y=x^{\lambda} \) is a solution of \[ y^{\prime \prime}-\frac{7}{x} y^{\prime}+\frac{16}{x^{2}} y=0 \] are: (a) \( \lambda=1,-1 \) (b) \( \lambda=8,2 \) (c) \( \lambda=4 \) (d) \( \lambda=-4 \) (e) None of the above.