Explain how to find the general solution to the given ODE.
\[ -40 \, e^{\left(4 \, t\right)} \sin\left(5 \, t\right) + 4 \, {y'} = 16 \, {y} \]
Answer:
\[ {y} = k e^{\left(4 \, t\right)} - 2 \, \cos\left(5 \, t\right) e^{\left(4 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ -24 \, e^{\left(2 \, t\right)} = 20 \, {y} - 4 \, {y'} \]
Answer:
\[ {y} = k e^{\left(5 \, t\right)} - 2 \, e^{\left(2 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 12 \, \cos\left(2 \, t\right) e^{\left(-t\right)} - 3 \, {y'} = 3 \, {y} \]
Answer:
\[ {y} = k e^{\left(-t\right)} + 2 \, e^{\left(-t\right)} \sin\left(2 \, t\right) \]
Explain how to find the general solution to the given ODE.
\[ -4 \, {y'} + 8 \, e^{\left(-5 \, t\right)} = 16 \, {y} \]
Answer:
\[ {y} = k e^{\left(-4 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 0 = -{y'} - {y} + 10 \, e^{\left(4 \, t\right)} \]
Answer:
\[ {y} = k e^{\left(-t\right)} + 2 \, e^{\left(4 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ {y} = 3 \, \cos\left(-t\right) e^{t} + {y'} \]
Answer:
\[ {y} = k e^{t} + 3 \, e^{t} \sin\left(-t\right) \]
Explain how to find the general solution to the given ODE.
\[ 4 \, {y'} - 16 \, {y} + 64 \, e^{\left(-4 \, t\right)} = 0 \]
Answer:
\[ {y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(-4 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 0 = 2 \, {y'} + 10 \, {y} - 4 \, e^{\left(-5 \, t\right)} \]
Answer:
\[ {y} = k e^{\left(-5 \, t\right)} + 2 \, t e^{\left(-5 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 0 = -24 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right) + 15 \, {y} + 3 \, {y'} \]
Answer:
\[ {y} = k e^{\left(-5 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 10 \, {y} - 2 \, {y'} = 6 \, e^{\left(5 \, t\right)} \]
Answer:
\[ {y} = k e^{\left(5 \, t\right)} - 3 \, t e^{\left(5 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 3 \, {y'} + 6 \, e^{t} = 3 \, {y} \]
Answer:
\[ {y} = k e^{t} - 2 \, t e^{t} \]
Explain how to find the general solution to the given ODE.
\[ -{y'} - {y} = -15 \, e^{\left(-t\right)} \sin\left(5 \, t\right) \]
Answer:
\[ {y} = k e^{\left(-t\right)} - 3 \, \cos\left(5 \, t\right) e^{\left(-t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 2 \, \cos\left(-t\right) e^{\left(3 \, t\right)} + {y'} = 3 \, {y} \]
Answer:
\[ {y} = k e^{\left(3 \, t\right)} + 2 \, e^{\left(3 \, t\right)} \sin\left(-t\right) \]
Explain how to find the general solution to the given ODE.
\[ 6 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right) + {y'} = 5 \, {y} \]
Answer:
\[ {y} = k e^{\left(5 \, t\right)} + 3 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 0 = -6 \, {y} - 2 \, {y'} - 30 \, e^{\left(2 \, t\right)} \]
Answer:
\[ {y} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(2 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 0 = 15 \, \cos\left(5 \, t\right) e^{\left(-2 \, t\right)} - 2 \, {y} - {y'} \]
Answer:
\[ {y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(5 \, t\right) \]
Explain how to find the general solution to the given ODE.
\[ 12 \, e^{\left(-4 \, t\right)} \sin\left(4 \, t\right) + 4 \, {y} + {y'} = 0 \]
Answer:
\[ {y} = k e^{\left(-4 \, t\right)} + 3 \, \cos\left(4 \, t\right) e^{\left(-4 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ -20 \, \cos\left(5 \, t\right) e^{\left(-5 \, t\right)} + 10 \, {y} = -2 \, {y'} \]
Answer:
\[ {y} = k e^{\left(-5 \, t\right)} + 2 \, e^{\left(-5 \, t\right)} \sin\left(5 \, t\right) \]
Explain how to find the general solution to the given ODE.
\[ -10 \, \cos\left(5 \, t\right) e^{\left(-4 \, t\right)} + {y'} + 4 \, {y} = 0 \]
Answer:
\[ {y} = k e^{\left(-4 \, t\right)} + 2 \, e^{\left(-4 \, t\right)} \sin\left(5 \, t\right) \]
Explain how to find the general solution to the given ODE.
\[ 2 \, {y'} = -6 \, {y} - 42 \, e^{\left(4 \, t\right)} \]
Answer:
\[ {y} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(4 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ -3 \, {y'} = 15 \, {y} + 9 \, e^{\left(-5 \, t\right)} \]
Answer:
\[ {y} = k e^{\left(-5 \, t\right)} - 3 \, t e^{\left(-5 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ -8 \, {y} = -24 \, e^{\left(2 \, t\right)} \sin\left(3 \, t\right) - 4 \, {y'} \]
Answer:
\[ {y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(3 \, t\right) e^{\left(2 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ -6 \, e^{\left(-4 \, t\right)} \sin\left(2 \, t\right) + 4 \, {y} = -{y'} \]
Answer:
\[ {y} = k e^{\left(-4 \, t\right)} - 3 \, \cos\left(2 \, t\right) e^{\left(-4 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ -12 \, \cos\left(-4 \, t\right) e^{t} + {y} - {y'} = 0 \]
Answer:
\[ {y} = k e^{t} + 3 \, e^{t} \sin\left(-4 \, t\right) \]
Explain how to find the general solution to the given ODE.
\[ 0 = -16 \, {y} + 4 \, {y'} + 12 \, e^{\left(4 \, t\right)} \]
Answer:
\[ {y} = k e^{\left(4 \, t\right)} - 3 \, t e^{\left(4 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ -36 \, \cos\left(4 \, t\right) e^{\left(5 \, t\right)} + 3 \, {y'} - 15 \, {y} = 0 \]
Answer:
\[ {y} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(4 \, t\right) \]
Explain how to find the general solution to the given ODE.
\[ 24 \, \cos\left(-4 \, t\right) e^{\left(2 \, t\right)} - 3 \, {y'} + 6 \, {y} = 0 \]
Answer:
\[ {y} = k e^{\left(2 \, t\right)} - 2 \, e^{\left(2 \, t\right)} \sin\left(-4 \, t\right) \]
Explain how to find the general solution to the given ODE.
\[ -54 \, e^{\left(4 \, t\right)} = 6 \, {y} + 3 \, {y'} \]
Answer:
\[ {y} = k e^{\left(-2 \, t\right)} - 3 \, e^{\left(4 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 12 \, {y} = 40 \, e^{\left(3 \, t\right)} \sin\left(-5 \, t\right) + 4 \, {y'} \]
Answer:
\[ {y} = k e^{\left(3 \, t\right)} - 2 \, \cos\left(-5 \, t\right) e^{\left(3 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ -5 \, {y} = -4 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right) - {y'} \]
Answer:
\[ {y} = k e^{\left(5 \, t\right)} + 2 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 2 \, {y'} = 4 \, e^{\left(-2 \, t\right)} \sin\left(-t\right) - 4 \, {y} \]
Answer:
\[ {y} = k e^{\left(-2 \, t\right)} + 2 \, \cos\left(-t\right) e^{\left(-2 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ -24 \, \cos\left(2 \, t\right) e^{\left(-4 \, t\right)} + 16 \, {y} = -4 \, {y'} \]
Answer:
\[ {y} = k e^{\left(-4 \, t\right)} + 3 \, e^{\left(-4 \, t\right)} \sin\left(2 \, t\right) \]
Explain how to find the general solution to the given ODE.
\[ 4 \, {y'} - 12 \, {y} - 16 \, e^{\left(5 \, t\right)} = 0 \]
Answer:
\[ {y} = k e^{\left(3 \, t\right)} + 2 \, e^{\left(5 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ -4 \, e^{t} = -4 \, {y} + 2 \, {y'} \]
Answer:
\[ {y} = k e^{\left(2 \, t\right)} + 2 \, e^{t} \]
Explain how to find the general solution to the given ODE.
\[ -8 \, {y} + 4 \, {y'} = 12 \, e^{\left(3 \, t\right)} \]
Answer:
\[ {y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(3 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ -{y} = -2 \, e^{\left(-t\right)} \sin\left(t\right) + {y'} \]
Answer:
\[ {y} = k e^{\left(-t\right)} - 2 \, \cos\left(t\right) e^{\left(-t\right)} \]
Explain how to find the general solution to the given ODE.
\[ -12 \, \cos\left(2 \, t\right) e^{\left(4 \, t\right)} - 3 \, {y'} = -12 \, {y} \]
Answer:
\[ {y} = k e^{\left(4 \, t\right)} - 2 \, e^{\left(4 \, t\right)} \sin\left(2 \, t\right) \]
Explain how to find the general solution to the given ODE.
\[ -4 \, {y'} = 4 \, {y} - 8 \, e^{\left(-t\right)} \]
Answer:
\[ {y} = k e^{\left(-t\right)} + 2 \, t e^{\left(-t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 6 \, {y} + 2 \, {y'} = 12 \, \cos\left(2 \, t\right) e^{\left(-3 \, t\right)} \]
Answer:
\[ {y} = k e^{\left(-3 \, t\right)} + 3 \, e^{\left(-3 \, t\right)} \sin\left(2 \, t\right) \]
Explain how to find the general solution to the given ODE.
\[ -16 \, \cos\left(2 \, t\right) e^{\left(-4 \, t\right)} = -16 \, {y} - 4 \, {y'} \]
Answer:
\[ {y} = k e^{\left(-4 \, t\right)} + 2 \, e^{\left(-4 \, t\right)} \sin\left(2 \, t\right) \]
Explain how to find the general solution to the given ODE.
\[ -4 \, {y'} = -32 \, e^{t} \sin\left(-4 \, t\right) - 4 \, {y} \]
Answer:
\[ {y} = k e^{t} + 2 \, \cos\left(-4 \, t\right) e^{t} \]
Explain how to find the general solution to the given ODE.
\[ 0 = -2 \, {y'} - 4 \, {y} + 6 \, e^{\left(-2 \, t\right)} \]
Answer:
\[ {y} = k e^{\left(-2 \, t\right)} + 3 \, t e^{\left(-2 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 3 \, {y'} = 12 \, {y} - 45 \, e^{\left(-t\right)} \]
Answer:
\[ {y} = k e^{\left(4 \, t\right)} + 3 \, e^{\left(-t\right)} \]
Explain how to find the general solution to the given ODE.
\[ -2 \, {y} = 2 \, {y'} - 4 \, e^{\left(-t\right)} \]
Answer:
\[ {y} = k e^{\left(-t\right)} + 2 \, t e^{\left(-t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 48 \, e^{\left(5 \, t\right)} = -3 \, {y'} - 9 \, {y} \]
Answer:
\[ {y} = k e^{\left(-3 \, t\right)} - 2 \, e^{\left(5 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 6 \, e^{\left(3 \, t\right)} = -3 \, {y'} + 9 \, {y} \]
Answer:
\[ {y} = k e^{\left(3 \, t\right)} - 2 \, t e^{\left(3 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 0 = 36 \, e^{\left(2 \, t\right)} \sin\left(-3 \, t\right) + 8 \, {y} - 4 \, {y'} \]
Answer:
\[ {y} = k e^{\left(2 \, t\right)} + 3 \, \cos\left(-3 \, t\right) e^{\left(2 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 2 \, {y'} + 2 \, {y} = 12 \, \cos\left(-2 \, t\right) e^{\left(-t\right)} \]
Answer:
\[ {y} = k e^{\left(-t\right)} - 3 \, e^{\left(-t\right)} \sin\left(-2 \, t\right) \]
Explain how to find the general solution to the given ODE.
\[ 3 \, {y} = 10 \, e^{\left(-3 \, t\right)} \sin\left(5 \, t\right) - {y'} \]
Answer:
\[ {y} = k e^{\left(-3 \, t\right)} - 2 \, \cos\left(5 \, t\right) e^{\left(-3 \, t\right)} \]
Explain how to find the general solution to the given ODE.
\[ 4 \, e^{\left(-5 \, t\right)} = -10 \, {y} - 2 \, {y'} \]
Answer:
\[ {y} = k e^{\left(-5 \, t\right)} - 2 \, t e^{\left(-5 \, t\right)} \]