3.4E: Exercises for Section 3.4

Exercise \(\PageIndex{1}\)

Find the general solution of \( 2y'' + 2y' - 4y = 0 \).

Exercise \(\PageIndex{2}\)

Find the general solution of \(y'' + 9y' - 10y = 0 \).

Exercise \(\PageIndex{3}\)

Solve \( y'' - 8y' + 16y = 0 \) for \(y(0) = 2, y'(0) = 0 \).

Exercise \(\PageIndex{4}\)

Solve \(y'' + 9y' = 0\) for \( y(0) = 1, y'(0) = 1\).

Exercise \(\PageIndex{5}\)

Find the general solution of \( 2y'' + 50y = 0 \).

Exercise \(\PageIndex{6}\)

Find the general solution of \( y'' + 6y' + 13y = 0 \).

Exercise \(\PageIndex{7}\)

Find the general solution of \( y'' = 0 \) using the methods of this section.

Exercise \(\PageIndex{8}\)

The method of this section applies to equations of other orders than two. We will see higher orders later. Try to solve the first order equation \( 2y' + 3y = 0 \) using the methods of this section.

Exercise \(\PageIndex{9}\)

Suppose that \( {(b - a)}^2 - 4ac < 0 \). Find a formula for the general solution of \( ax^2y'' + bxy' + cy = 0 \).

Hint: Note that \( x^r = e^{r \ln x} \).

Exercise \(\PageIndex{10}\)

Find the solution to \(y''-(2\alpha )y'+\alpha^{2}y=0\), \(y(0)=a\), \(y'(0)=b\), where \(\alpha\), \(a\), and \(b\) are real numbers.

Exercise \(\PageIndex{11}\)

Construct an equation such that \(y=C_{1}e^{-2x}\cos (3x)+C_{2}e^{-2x}\sin (3x)\) is the general solution.

Exercise \(\PageIndex{12}\)

Find the general solution to \(y'' + 4y' + 2y = 0 \).

Answer

\(y=C_{1}e^{(-2+\sqrt{2})x}+C_{2}e^{(-2-\sqrt{2})x}\)

Exercise \(\PageIndex{13}\)

Find the general solution to \( y'' - 6y' + 9y = 0\).

Answer

\(y=C_{1}e^{3x}+C_{2}xe^{3x}\)

Exercise \(\PageIndex{14}\)

Find the solution to \( 2y'' + y' + y = 0, y(0) = 1, y'(0) = -2 \).

Answer

\(y=e^{-x/4}\cos\left((\frac{\sqrt{7}}{4})x\right)-\sqrt{7}e^{-x/4}\sin\left((\frac{\sqrt{7}}{4})x\right)\)

Exercise \(\PageIndex{15}\)

Find the solution to \( 2y'' + y' - 3y = 0, y(0) = a, y'(0) = b \).

Answer

\(y=\frac{2(a-b)}{5}e^{-3x/2}+\frac{3a+2b}{5}e^{x}\)

Exercise \(\PageIndex{16}\)

Find the solution to \( z''(t) = -2z'(t) - 2z (t), z(0) = 2, z'(0) = -2 \).

Answer

\(z(t)=2e^{-t}\cos (t)\)

Exercise \(\PageIndex{17}\)

Find the solution to \(y''-(\alpha +\beta )y'+\alpha\beta y=0\), \(y(0)=a\), \(y'(0)=b\), where \(\alpha\), \(\beta\), \(a\), and \(b\) are real numbers, and \(\alpha\neq\beta\).

Answer

\(y=\frac{\alpha\beta -b}{\beta -\alpha}e^{\alpha x}+\frac{b-a\alpha}{\beta -\alpha}e^{\beta x}\)

Exercise \(\PageIndex{18}\)

Construct an equation such that \(y=C_{1}e^{3x}+C_{2}e^{-2x}\) is the general solution.

Answer

\(y''-y'-6y=0\)