Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 126 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 127 Higher Order DEs Solve the nth degree polynomial: If the roots are all real & distinct, Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 128 If the root m1 is of multiplicity k, the linear independent solutions are: And the general solution is a linear combination of the above. Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 129 Example 1.17 Solve: Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 130 Undetermined Coefficients To Solve: We Must: - Solve the Find the complimentary solution -Find any particular solution of the nonhomogeneous equation Assume: Fall 2021 has the same form as Applied Mathematics Lecture 01: Ordinary Differential Equations 131 Example 1.18 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 132 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 133 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 134 Example 1.19 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 135 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 136 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 137 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 138 Variation of Parameters Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 139 If y1 and y2 are linearly independent: Variation of parameters: Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 140 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 141 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 142 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 143 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 144 Example 1.20 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 145 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 146 Example 1.21 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 147 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 148 Cauchy-Euler Equations Trial Solution: Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 149 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 150 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 151 Distinct, Real Roots: Repeated Roots: Complex Roots: Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 152 Example 1.22 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 153 Example 1.23 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 154 Free Vibrations Image: Kreyszig Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 155 Hooke’s Law: Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 156 Define: Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 157 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 158 Period: Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 159 Alternate Form Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 160 Trig Identities Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 161 Image: ZIll Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 162 Image: Zill Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 163 Example 1.24 An object weighing 2 lb stretches a spring 6 inches. At t = 0, the object is released from a point 8 inches below the equilibrium position. Determine the equation of motion for the object if its initial velocity is (a) 4/3 ft/s upward; (b) 0; (c) 4/3 ft/s downward. Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 164 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 165 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 166 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 167 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 168 Damped Vibrations Image: Zill Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 169 Recall: Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 170 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 171 Case I: Overdamping Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 172 Case II: Critical Damping Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 173 Example 1.25 An 8 lb weight stretches a spring 2 feet. The damping coefficient is twice the instantaneous velocity. The weight is released from its equilibrium position with an upward velocity of 3 ft/s. Determine the equation of motion for the object. Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 174 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 175 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 176 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 177 Case III: Underdamping Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 178 Example 1.26 An 16 lb weight is attached to a 5 foot spring. At equilibrium, the spring measures 8.2 feet. If the weight is pushed up and released from rest at a point 2 feet above the equilibrium position, find the equation of motion for the object ( c = 1lb-s/ft). Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 179 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 180 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 181 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 182 Beam Deflection Image: Zill Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 183 Image: Zill Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 184 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 185 Boundary Conditions Image: Zill Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 186 Example 1.27 A beam of length 1 m is fixed at both ends. Find the deflection of the beam if a constant load is uniformly distributed along its’ length. Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 187 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 188 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 189 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 190 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 191