7.4 Exercises
Your Turn
Is the power series ∑ k = 0 ∞ e k x k \sum_{k=0}^\infty e^k x^k
Your Turn
Is the power series ∑ k = 0 ∞ k x k \sum_{k=0}^\infty k x^k
Your Turn
Is the power series ∑ k = 0 ∞ k ! x k \sum_{k=0}^\infty k! x^k
Your Turn
Is the power series ∑ k = 0 ∞ 1 ( 2 k ) ! ( x − 10 ) k \sum_{k=0}^\infty \frac{1}{(2k)!} {(x-10)}^k
Your Turn
Determine the Taylor series for ln x \ln x x 0 = 1 x_0 = 1
Your Turn
Determine the Taylor series and its radius of convergence of x 4 − x 2 \dfrac{x}{4-x^2} x 0 = 0 x_0 = 0
Your Turn
Suppose that the ratio test applies to a series ∑ k = 0 ∞ a k x k \sum_{k=0}^\infty a_k x^k
Your Turn
Is the power series ∑ n = 1 ∞ ( 0.1 ) n x n \sum_{n=1}^\infty {(0.1)}^n x^n
Answer
Yes. Radius of convergence is 10 10
Your Turn
(challenging)
Is the power series ∑ n = 1 ∞ n ! n n x n \sum_{n=1}^\infty \frac{n!}{n^n} x^n
Answer
Yes. Radius of convergence is e e
Your Turn
Using the geometric series, expand 1 1 − x \frac{1}{1-x} x 0 = 2 x_0=2 x x
Answer
1 1 − x = − 1 1 − ( 2 − x ) \frac{1}{1-x}=-\frac{1}{1-(2-x)} 1 1 − x = ∑ n = 0 ∞ ( − 1 ) n + 1 ( x − 2 ) n \frac{1}{1-x}=\sum\limits_{n=0}^\infty (-1)^{n+1}(x-2)^{n} 1 < x < 3 1<x<3
Your Turn
(challenging)
Find the Taylor series for x 7 e x x^7 e^x x 0 = 0 x_0 = 0
Answer
∑
n
=
7
∞
1
(
n
−
7
)
!
x
n
\sum\limits_{n=7}^\infty \frac{1}{(n-7)!} x^{n}
Your Turn
(challenging)
Imagine f f g g f ( k ) ( 0 ) = g ( k ) ( 0 ) f^{(k)}(0) = g^{(k)}(0) k k f ( x ) − g ( x ) f(x)-g(x)
Answer
f ( x ) − g ( x ) f(x)-g(x)
In the following exercises, when asked to solve an equation using power series methods, you should find the first few terms of the series, and if possible find a general formula for the k th k^{\text{th}}
Your Turn
The methods work for other orders than second order. Try the methods of this section to solve the first order system y ′ − x y = 0 y'-xy = 0 x 0 = 0 x_0 = 0
Your Turn
Use power series methods to solve ( 1 − x ) y ″ + y = 0 (1-x)y''+y = 0 x 0 = 0 x_0 = 0 Use the solution to part a) to find a solution for x y ″ + y = 0 xy''+y=0 x 0 = 1 x_0=1 Your Turn
Use power series methods to solve y ″ + 2 x 3 y = 0 y'' + 2 x^3 y = 0 x 0 = 0 x_0 =0
Answer
a 2 = 0 a_{2}=0 a 3 = 0 a_{3}=0 a 4 = 0 a_{4}=0 k ≥ 5 k\geq 5 a k = − 2 a k − 5 k ( k − 1 ) a_{k}=\frac{-2a_{k-5}}{k(k-1)} y ( x ) = a 0 + a 1 x − a 0 10 x 5 − a 1 15 x 6 + a 0 450 x 10 + a 1 825 x 11 − a 0 47250 x 15 − a 1 99000 x 16 + ⋯ y(x)=a_{0}+a_{1}x-\frac{a_{0}}{10}x^{5}-\frac{a_{1}}{15}x^{6}+\frac{a_{0}}{450}x^{10}+\frac{a_{1}}{825}x^{11}-\frac{a_{0}}{47250}x^{15}-\frac{a_{1}}{99000}x^{16}+\cdots
Your Turn
Attempt to solve x 2 y ″ − y = 0 x^2 y'' - y = 0 x 0 = 0 x_0 = 0 x 0 x_0
Answer
Applying the method of this section directly we obtain a k = 0 a_{k}=0 k k y ( x ) = 0 y(x)=0
Your Turn
In the following equations classify the point x = 0 x=0
x
2
(
1
+
x
2
)
y
″
+
x
y
=
0
x^2(1+x^2)y''+xy=0
x
2
y
″
+
y
′
+
y
=
0
x^2y''+y'+y=0
x
y
″
+
x
3
y
′
+
y
=
0
xy''+x^3y'+y=0
x
y
″
+
x
y
′
−
e
x
y
=
0
xy''+xy'-e^xy=0
x
2
y
″
+
x
2
y
′
+
x
2
y
=
0
x^2y''+x^2y'+x^2y=0
Your Turn
In the following equations classify the point x = 0 x=0
y
″
+
y
=
0
y''+y=0
x
3
y
″
+
(
1
+
x
)
y
=
0
x^3y''+(1+x)y=0
x
y
″
+
x
5
y
′
+
y
=
0
xy''+x^5y'+y=0
sin
(
x
)
y
″
−
y
=
0
\sin(x)y''-y=0
cos
(
x
)
y
″
−
sin
(
x
)
y
=
0
\cos(x)y''-\sin(x)y=0
Answer
ordinary, singular but not regular singular, regular singular, regular singular, ordinary. Your Turn
Find the general solution of x 2 y ″ − y = 0 x^2 y'' -y = 0
Answer
y
=
A
x
1
+
5
2
+
B
x
1
−
5
2
y=Ax^{\frac{1+\sqrt{5}}{2}}+Bx^{\frac{1-\sqrt{5}}{2}}
Your Turn
Find a particular solution of x 2 y ″ + ( x − 3 4 ) y = 0 x^2 y'' +(x-\frac{3}{4})y = 0
Answer
y = x 3 / 2 ∑ k = 0 ∞ ( − 1 ) − 1 k ! ( k + 2 ) ! x k y=x^{3/2}\sum\limits_{k=0}^\infty \frac{(-1)^{-1}}{k!(k+2)!}x^{k} a 0 = 1 a_{0}=1
Your Turn
(tricky)
Find the general solution of x 2 y ″ − x y ′ + y = 0 x^2 y'' - x y' +y = 0
Answer
y
=
A
x
+
B
x
ln
(
x
)
y=Ax+Bx\ln (x)
Adapted from Differential Equations for Engineers by Jiří Lebl (https://www.jirka.org/diffyqs/), © Jiří Lebl, licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0 .
