Home / AP Calculus BC: 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables – Exam Style questions with Answer- FRQ

AP Calculus BC: 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables – Exam Style questions with Answer- FRQ

Question

Let y = f(x) be the particular solution to the differential equation \(\frac{dy}{dx}=y\cdot (x In x)\) with initial condition f(1) = 4 . It can be shown that f”(1) = 4.
(a) Write the second-degree Taylor polynomial for f about x = 1. Use the Taylor polynomial to approximate f(2) .
(b) Use Euler’s method, starting at x = 1 with two steps of equal size, to approximate f(2) . Show the work that leads to your answer.
(c) Find the particular solution y = f(x) to the differential equation \(\frac{dy}{dx}=y\cdot (x In x)\) with initial condition f(1) = 4 . 

Answer/Explanation

Ans:

(a)

\(\frac{dy}{dx}= 4.1 In 1\)

\(P_{2}(x)=4+In 1 (x-1)+\frac{4(x-1)^{2}}{2!}\)                                \(P_{2}(x)=4+\frac{4(x-1)^{2}}{2!}\)

\(P_{2}(2)=4+\frac{4(2-1)^{2}}{2!}= 4+2 = 6\)

(b)

\(\frac{dy}{dx}= 4\cdot 1.5 In 1.5\)

6 In 1.5

f(2) = 4 + 3 In 1.5

(c)

\(\frac{dy}{dx}= y\cdot sin x\)

\(\frac{1}{y}dy= x In x dx\)                                                           u = In x                                               dv = x dx

\(\int \frac{1}{y}dy= \int x In x dx\)                                            \(du = \frac{1}{x}dx\)                  \(v = \frac{1}{x}x^{2}\)

\(In y = \frac{1}{2}x^{2}In x -\frac{1}{4}x^{2}+ c\)

\(In 4 = \frac{1}{2}In 1 -\frac{1}{4}+ c\)

\(In 4 + \frac{1}{4}= c\)

\(In y =\frac{1}{2}x^{2}In x- \frac{1}{4}x^{2}+In 4 + \frac{1}{4}\)
\(y =4e^{\frac{1}{2}x^{2}In x}- \frac{1}{4}x^{2} + \frac{1}{4}\)

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