## C1 - Homogeneous first-order linear IVP

#### Example 1 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$12 \, {y} - 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -32$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(3 \, t\right)}$

${y} = -4 \, e^{\left(3 \, t\right)}$

#### Example 2 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$9 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -24$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(3 \, t\right)}$

${y} = -3 \, e^{\left(3 \, t\right)}$

#### Example 3 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-9 \, {y} = 3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{1}{9}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-3 \, t\right)}$

${y} = -3 \, e^{\left(-3 \, t\right)}$

#### Example 4 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$12 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-3 \, t\right)}$

${y} = 2 \, e^{\left(-3 \, t\right)}$

#### Example 5 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 27$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(2 \, t\right)}$

${y} = 3 \, e^{\left(2 \, t\right)}$

#### Example 6 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$0 = -4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = -2 \, e^{\left(-2 \, t\right)}$

#### Example 7 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-10 \, {y} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{3}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = 3 \, e^{\left(-2 \, t\right)}$

#### Example 8 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$4 \, {y'} = -12 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{27}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-3 \, t\right)}$

${y} = 2 \, e^{\left(-3 \, t\right)}$

#### Example 9 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$0 = 8 \, {y} + 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = -2 \, e^{\left(-2 \, t\right)}$

#### Example 10 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$6 \, {y} = -3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = -2 \, e^{\left(-2 \, t\right)}$

#### Example 11 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-8 \, {y} + 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -8$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(2 \, t\right)}$

${y} = -2 \, e^{\left(2 \, t\right)}$

#### Example 12 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-3 \, {y'} = -9 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(3 \, t\right)}$

${y} = -4 \, e^{\left(3 \, t\right)}$

#### Example 13 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(2 \, t\right)}$

${y} = -4 \, e^{\left(2 \, t\right)}$

#### Example 14 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-8 \, {y} + 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 16$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(2 \, t\right)}$

${y} = 4 \, e^{\left(2 \, t\right)}$

#### Example 15 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-3 \, t\right)}$

${y} = -2 \, e^{\left(-3 \, t\right)}$

#### Example 16 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$12 \, {y} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -81$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(3 \, t\right)}$

${y} = -3 \, e^{\left(3 \, t\right)}$

#### Example 17 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-6 \, {y} = -3 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(2 \, t\right)}$

${y} = 2 \, e^{\left(2 \, t\right)}$

#### Example 18 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$0 = 2 \, {y'} + 4 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -1$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = -4 \, e^{\left(-2 \, t\right)}$

#### Example 19 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$0 = -6 \, {y} - 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-3 \, t\right)}$

${y} = 2 \, e^{\left(-3 \, t\right)}$

#### Example 20 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$3 \, {y'} = 9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(3 \, t\right)}$

${y} = -2 \, e^{\left(3 \, t\right)}$

#### Example 21 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$4 \, {y'} - 8 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -27$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(2 \, t\right)}$

${y} = -3 \, e^{\left(2 \, t\right)}$

#### Example 22 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$4 \, {y'} - 12 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -16$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(3 \, t\right)}$

${y} = -2 \, e^{\left(3 \, t\right)}$

#### Example 23 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$0 = 4 \, {y} - 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 12$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(2 \, t\right)}$

${y} = 3 \, e^{\left(2 \, t\right)}$

#### Example 24 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$2 \, {y'} = -6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{8}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-3 \, t\right)}$

${y} = 3 \, e^{\left(-3 \, t\right)}$

#### Example 25 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-3 \, {y'} + 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -24$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(3 \, t\right)}$

${y} = -3 \, e^{\left(3 \, t\right)}$

#### Example 26 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$6 \, {y} + 2 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-3 \, t\right)}$

${y} = -4 \, e^{\left(-3 \, t\right)}$

#### Example 27 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-8 \, {y} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{4}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = 3 \, e^{\left(-2 \, t\right)}$

#### Example 28 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$0 = 5 \, {y'} + 15 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-3 \, t\right)}$

${y} = -4 \, e^{\left(-3 \, t\right)}$

#### Example 29 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$5 \, {y'} - 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 8$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(2 \, t\right)}$

${y} = 2 \, e^{\left(2 \, t\right)}$

#### Example 30 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$2 \, {y'} + 6 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-3 \, t\right)}$

${y} = -4 \, e^{\left(-3 \, t\right)}$

#### Example 31 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$0 = -3 \, {y'} + 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -27$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(2 \, t\right)}$

${y} = -3 \, e^{\left(2 \, t\right)}$

#### Example 32 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = -2 \, e^{\left(-2 \, t\right)}$

#### Example 33 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$0 = -10 \, {y} - 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 1$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = 4 \, e^{\left(-2 \, t\right)}$

#### Example 34 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$6 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-3 \, t\right)}$

${y} = -2 \, e^{\left(-3 \, t\right)}$

#### Example 35 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$0 = 6 \, {y} + 3 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 1$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = 4 \, e^{\left(-2 \, t\right)}$

#### Example 36 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-6 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = 4 \, e^{\left(-2 \, t\right)}$

#### Example 37 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$0 = 4 \, {y'} - 12 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -54$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(3 \, t\right)}$

${y} = -2 \, e^{\left(3 \, t\right)}$

#### Example 38 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-8 \, {y} + 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -12$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(2 \, t\right)}$

${y} = -3 \, e^{\left(2 \, t\right)}$

#### Example 39 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = 4 \, e^{\left(-2 \, t\right)}$

#### Example 40 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$0 = 3 \, {y'} + 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{2}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = 2 \, e^{\left(-2 \, t\right)}$

#### Example 41 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = -2 \, e^{\left(-2 \, t\right)}$

#### Example 42 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 24$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(3 \, t\right)}$

${y} = 3 \, e^{\left(3 \, t\right)}$

#### Example 43 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-10 \, {y} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{3}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = 3 \, e^{\left(-2 \, t\right)}$

#### Example 44 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$0 = -5 \, {y'} + 10 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(2 \, t\right)}$

${y} = -4 \, e^{\left(2 \, t\right)}$

#### Example 45 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$9 \, {y} = 3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 81$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(3 \, t\right)}$

${y} = 3 \, e^{\left(3 \, t\right)}$

#### Example 46 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-5 \, {y'} - 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -1$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = -4 \, e^{\left(-2 \, t\right)}$

#### Example 47 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$12 \, {y} - 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 32$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(3 \, t\right)}$

${y} = 4 \, e^{\left(3 \, t\right)}$

#### Example 48 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-3 \, t\right)}$

${y} = -3 \, e^{\left(-3 \, t\right)}$

#### Example 49 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-10 \, {y} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$

${y} = 2 \, e^{\left(-2 \, t\right)}$

#### Example 50 π

Explain how to find the general solution to the given ODE, and the particular solution to the given IVP.

$-4 \, {y'} = 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -1$

Then show how to verify that your particular solution is correct.

${y} = k e^{\left(-2 \, t\right)}$
${y} = -4 \, e^{\left(-2 \, t\right)}$