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Variation of parameters Finding a particular solution when a homogeneous solution appears in the right-side function is handled using a method called variation of parameters. (This method is also used to find particular solutions of equations that cannot be handled by undetermined coefficients.) The following steps show how variation of parameters is used to find the particular solution of one specific equation.
a. Consider the equation y" – y – et. Show that the homogeneous solutions are y1 = et and y2 = e–t. Note that the right–side function is a homogeneous solution.
b. Assume a particular solution has the form
where the functions u1 and u2 are to be determined. Compute yp' and impose the condition u1'et + u2'e–t = 0 to show that ypʹ = u1et – u2′e–t.
c. Compute yp" and substitute it into the
d. Parts (c) and (d) give two equations for u1ʹ and u2ʹ Eliminate u2ʹ and show that the equation for u1 is
e. Solve the equation in part (d) for u1.
f. Use part (e) to show that the equation for u2 is
g. Solve the equation in part (f) for u2.
h. Now assemble the particular solution yp(t) =u1(t)et + u2(t)e–t and show that
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Chapter D2 Solutions
Student Solutions Manual, Single Variable for Calculus: Early Transcendentals
- 3. Solve the equation, give the answer exactly (no calculator approximations), and show all your work. (4 points) log5 2x = 3arrow_forwardLet I = f(x) dx, where f is the function whose graph is shown. 4 2 y f X 1 2 3 4 (a) Use the graph to find L2, R2 and M2. R₂ M2 = = = (b) Are these underestimates or overestimates of I? O 42 is an underestimate. O 42 is an overestimate. ◇ R2 is an underestimate. OR2 is an overestimate. OM2 is an underestimate. ○ M2 is an overestimate. (c) Use the graph to find T2. T₂ =arrow_forwardVector u has a magnitude of 23 and vector v has a magnitude of 83. The angle between the two vectors is 126 degrees.a) Draw a fully-labelled vector diagram showing the two vectors and the resultant vector when they are added together.b) Find the magnitude of the resultant vector.c) Find the direction of the resultant vector relative to vector u. Solding by finding the x and y of the vectors and addingarrow_forward
- 3) If a is a positive number, what is the value of the following double integral? 2a Love Lv 2ay-y² .x2 + y2 dadyarrow_forward16. Solve each of the following equations for x. (a) 42x+1 = 64 (b) 27-3815 (c) 92. 27² = 3-1 (d) log x + log(x - 21) = 2 (e) 3 = 14 (f) 2x+1 = 51-2xarrow_forward11. Find the composition fog and gof for the following functions. 2 (a) f(x) = 2x+5, g(x) = x² 2 (b) f(x) = x²+x, g(x) = √√x 1 (c) f(x) = -1/2) 9 9(x) = х = - Xarrow_forward
- practice problem please help!arrow_forward13. A restaurant will serve a banquet at a cost of $20 per person for the first 50 people and $15 for person for each additional person. (a) Find a function C giving the cost of the banquet depending on the number of people p attending. (b) How many people can attend the banquet for $2000?arrow_forwardAlt Fn Ctrl 12. Find functions f and g such that h(x) = (fog)(x). (a) h(x) = (x² + 2)² x+1 (b) h(x) = 5 3arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageGlencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill
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