Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 4 - Vector Spaces - 4.8 Applications of Vector Spaces - 4.8 Exercises - Page 219: 22


$$W=-4 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{6}+2 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{5}+6 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{4}-2 \mathrm{e}^{x} x^{3} \mathrm{e}^{x^{2}}-6 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{2}.$$

Work Step by Step

The Wronskian of $\left\{x^{2}, e^{x^{2}}, x^{2} e^{x}\right\}$ is given by $$W=\left[\begin{array}{lll}{x^{2}} & {\mathrm{e}^{x^{2}}} & {\mathrm{e}^{x} x^{2}} \\ {2 x} & {2 x \mathrm{e}^{x^{2}}} & {\mathrm{e}^{x} x^{2}+2 x \mathrm{e}^{x}} \\ {2} & {2 \mathrm{e}^{x^{2}}+4 x^{2} \mathrm{e}^{x^{2}} }&{ \mathrm{e}^{x} x^{2}+4 x \mathrm{e}^{x}+2 \mathrm{e}^{x}}\end{array}\right|\\=-4 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{6}+2 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{5}+6 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{4}-2 \mathrm{e}^{x} x^{3} \mathrm{e}^{x^{2}}-6 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{2}.$$
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