Hence the solution of the initial value problem is

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Unformatted text preview: ar singular point. ________________________________________________________________________ page 211 —————————————————————————— CHAPTER 5. —— Section 5.5 1. Substitution of The roots are results in the quadratic equation , 3. Substitution of The root is , is results in the quadratic equation , where , with multiplicity two . Hence the general solution, for 5. Substitution of The root is . Hence the general solution, for , where results in the quadratic equation , is , where , with multiplicity two . Hence the general solution, for 6. Substitution of The roots are 7. Substitution of The roots are 8. Substitution of results in the quadratic equation , . Hence the general solution, for , where , is results in the quadratic equation . Hence the general solution, for results in the quadratic equation , is , where , is , where ________________________________________________________________________ page 212 —————————————————————————— CHAPTER 5. —— The roots are complex, with . Hence the general solution, for , is 10. Substitution of results in the quadratic equation The roots are complex, with 11. Substitution of . Hence the general solution, for results in the quadratic equation The roots are complex, with , where , is , where . Hence the general solution, for , is 12. Substitution of The roots are 14. Substitution of results in the quadratic equation , . Hence the general solution, for , where , is results in the quadratic equation The roots are complex, with , is , where . Hence the general solution, for Invoking the initial conditions, we obtain the system of equations ________________________________________________________________________ page 213 —————————————————————————— CHAPTER 5. —— Hence the solution of the initial value problem is As , the solution decreases without bound. 15. Substitution of The root is results in the quadratic equation , where , with multiplicity two . Hence the general solution, for , is Invoking the initial conditions, we obtain the system of equations Hence the solution of the initial value problem is ________________________________________________________________________ page 214 —————————————————————————— CHAPTER 5. —— We find that 18. 36 as . Substitution of results in the quadratic equation . The roots are . If , the roots are complex, with solution, for , is . Hence the general Since the trigonometric factors are bounded, are equal, and Since lim , as as . If . If , the roots , the roots are real, with . Hence the general solution, for Evidently, solutions approach zero as long as , is That is, . Hence all solutions approach zero, for 1 379. are . Substitution of results in the quadratic equation , . Hence the general solution, for , is . The roots Invoking the initial conditions, we obtain the system of equations ________________________________________________________________________ p...
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