1. Math 3C Practice Midterm #1 Solutions Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

2. 1. Consider the following differential equation: a) Sketch a slope field for this differential equation for values of t between -2 and 2, and values of y between -2 and 2. d) Sketch a solution to this differential equation with the initial condition y(0)=0. y(0)=0 solution y=1 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

3. 1. Consider the following differential equation: b) What are the constant solutions to this differential equation, if any? c) Classify any constant solutions as stable or unstable. Constant solutions happen when y=constant and y=1 is a constant solution To see if it is stable, check the sign of the derivative on either side: This is negative when t>0, so the solution curve will drop toward y=1 This is positive when t>0. so the solution curve will rise toward y=1 This is a STABLE solution Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

4. 1. Consider the following differential equation: e) Is this a separable differential equation? If so, find the general solution using separation of variables, and find the solution with the initial condition y(0)=0. Yes, this is separable: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

5. 1. Consider the following differential equation: f) Is this a linear differential equation? If so, find the general solution by finding the homogeneous solution, then the particular solution, and adding the two. This is linear – the more standard form is: We’ll use the Euler-Lagrange method. The Homogeneous solution is: Now, for the particular solution we have to solve for v: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

6. 2) Consider the following differential equation: a) Use Euler’s method to find y(1) with step sizes of 0.5 and 0.25. Euler’s method uses the initial condition as a starting point, then calculates the slope from the original diff. eq. and uses that slope to calculate the next y value. A table of values is shown below. The graph shows the exact solution, along with the Euler approximations. Detailed calculations for the n=4 case are also shown. ty (n=4)y' (n=4)y (n=2)y' (n=2) 0.001.0000.00010 0.251.000-0.250 0.500.938-0.5331-0.5 0.750.804-0.933 1.000.571 0.75 When n=4 the step size is h=0.25, so we will calculate y values at t=0.25,0.5,0.75 and 1.0 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

7. 2) Consider the following differential equation: b) Find the exact value for y(1) by solving the differential equation. c) Is this a linear differential equation? Homogeneous? Constant coefficient? This diff. eq. is nonlinear, so the concepts of homogeneity and constant coefficient do not apply. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

8. 3) Classify the following differential equations as linear, separable, both, or neither. If the differential equation is linear, further classify it as homogeneous or non-homogeneous, and constant coefficient or variable coefficient. If the differential equation is either linear or separable, then find the general solution. a) b) c) d) Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

9. 3) Classify the following differential equations as linear, separable, both, or neither. If the differential equation is linear, further classify it as homogeneous or non-homogeneous, and constant coefficient or variable coefficient. If the differential equation is either linear or separable, then find the general solution. a) This equation is linear, but not separable (try separating it – I dare you!) It is non-homogeneous (the right hand side is not 0), and has constant coefficients. Here is the solution using 2 different methods: INTEGRATING FACTOR UNDETERMINED COEFFICIENTS Constant coeffeicent, so the factor is easy: Multiply through: Notice that this is an exact differential: Integrate both sides: Divide: Simplify: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

10. 3) Classify the following differential equations as linear, separable, both, or neither. If the differential equation is linear, further classify it as homogeneous or non-homogeneous, and constant coefficient or variable coefficient. If the differential equation is either linear or separable, then find the general solution. b) This is nonlinear, but it is separable: Integrate both sides: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

11. 3) Classify the following differential equations as linear, separable, both, or neither. If the differential equation is linear, further classify it as homogeneous or non-homogeneous, and constant coefficient or variable coefficient. If the differential equation is either linear or separable, then find the general solution. c) Nonlinear and not separable. We don’t have an obvious method for this one. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

12. 3) Classify the following differential equations as linear, separable, both, or neither. If the differential equation is linear, further classify it as homogeneous or non-homogeneous, and constant coefficient or variable coefficient. If the differential equation is either linear or separable, then find the general solution. d) This is linear, separable, homogeneous and variable coefficient. We could use separation, or an integrating factor. SEPARATIONINTEGRATING FACTOR Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

13. 4) Use the change of variable z = ln(y) to solve the non-linear differential equation (assume a and b are constants) Substitute into the diff. eq.: This is linear, with constant coefficients. Let’s use an integrating factor: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB