Differential Equation Questions Worksheet

Differential Equation Questions Worksheet

International Association University Summer Sessions (IAUSS) Assignment 1 Differential Equation Summer Session 2020 Mr. Hernández Student’s Name: 1) State the type and order of the given differential equation. Determinate whether or not the equation is linear, if nonlinear explain why? Identify the dependent and independent variables. a) 𝑥 b) 𝑑3 𝑦 𝑑𝑦 4 𝑑𝑥 𝑑𝑥 −( ) +𝑦 = 0 3 𝜕4 𝑢 𝜕2 𝑢 𝜕𝑥 𝜕𝑦 2 +5 4 − 0.33𝑢 = 0 2) Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution: a) 𝑦 ′′ − 6𝑦 ′ + 13𝑦 = 0; 𝑦 = 𝑒 3𝑥 cos 2𝑥 3) Verify that the indicated expression is an implicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution: a) 𝑑𝑋 𝑑𝑡 = (𝑋 − 1)(1 − 2𝑋); 2𝑋−1 ln ( 𝑋−1 ) = 𝑡 4) Verify that the function 𝜑(𝑥 ) = 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −2𝑥 is a solution to the linear equation 𝑑2 𝑦 𝑑𝑥 2 𝑑𝑦 + 𝑑𝑥 − 2𝑦 = 0, for any choice of the constants 𝒄𝟏 and 𝒄𝟐 . Determine 𝒄𝟏 and 𝒄𝟐 so that the initial condition 𝑦(0) = 2, 𝑦 ′ (0) = 1 is satisfied. 5) Solve the initial value problem using “separable variable” 𝑑𝑦 𝑑𝑥 𝑦−1 = 𝑥+3 ; 𝑦(−1) = 0 6) Give an example of a Homogenous and a nonhomogeneous linear differential equation. 1 of 2 7) Heart Pacemaker consists of a switch, battery of constant voltage 𝐸0 , a capacitor with constant capacitance 𝐶, and the heart as a resistor with constant resistance R. When the switch is closed, the capacitor charges; when the switch is open, the capacitor discharges, sending an electrical stimulus to the heart. During the time the heart is being stimulated, the voltage 𝐸 across the heart satisfies the linear differential equation 𝑑𝐸 1 =− 𝐸. 𝑑𝑡 𝑅𝐶 Solve the DE subject to 𝐸 (4) = 𝐸0 . 8) In the following exercises determine whether the given D.E. is exact. If it’s exact, solve it. If not, find the multiple for exactness and then solve it. a) (2𝑦 − 6𝑥 )𝑑𝑥 + (3𝑥 + 4𝑥 2 𝑦 −1 )𝑑𝑦 = 0 𝑑𝑦 b) 𝑥 𝑑𝑥 = 2𝑥𝑒 2 − 𝑦 + 6𝑥 2 9) In 1790 the population of the United States was 3.93 million, and in 1890 it was 62.98 million. Using the Malthusian model [𝑝(𝑡) = 𝑝0 𝑒 𝑘𝑡 ], estimate the U.S. population as a function of time. Use your result to predict the population in 1900. 2 of 2 …
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