17.5E: Exercises for Section 17.4

Find a power series solution for each of the following differential equations.

1. \(y″+6y′=0\)

2. \(5y″+y′=0\)

Answer

\(\displaystyle y=c_0+5c_1 \sum_{n=1}^∞ \frac{(−x/5)^n}{n!}=c_0+5c_1e^{−x/5}\)

3. \(y''+25y=0\)

4. \(y''−y=0\)

Answer

\(\displaystyle y=c_0 \sum_{n=0}^∞ \frac{(x)^{2n}}{(2n)!}+c_1 \sum_{n=0}^∞ \frac{(x)^{2n+1}}{(2n+1)!}\)

5. \(2y′+y=0\)

6. \(y′−2xy=0\)

Answer

\(\displaystyle y=c_0 \sum_{n=0}^∞ \frac{x^{2n}}{n!}=c_0e^{x^2}\)

7. \((x−7)y′+2y=0\)

8. \(y''−xy′−y=0\)

Answer

\(\displaystyle y=c_0 \sum_{n=0}^∞ \frac{x^{2n}}{2^nn!}+c_1 \sum_{n=0}^∞ \frac{x^{2n+1}}{1⋅3⋅5⋅7⋯(2n+1)}\)

9. \((1+x^2)y''−4xy′+6y=0\)

10. \(x^2y''−xy′−3y=0\)

Answer

\(y=c_1x^3+\dfrac{c_2}{x}\)

11. \(y″−8y′=0, \quad y(0)=−2, \; y′(0)=10\)

12. \(y″−2xy=0, \quad y(0)=1, \; y′(0)=−3\)

Answer

\(y=1−3x+\dfrac{2x^3}{3!}−\dfrac{12x^4}{4!}+\dfrac{16x^6}{6!}−\dfrac{120x^7}{7!}+⋯\)

13. The differential equation \(x^2y″+xy′+(x^2−1)y=0\) is a Bessel equation of order \(1.\) Use a power series of the form \(\displaystyle y=\sum_{n=0}^∞ a_nx^n\) to find the solution.