Solve the Differential Equation $\frac { d ^ { 2 } y } { d t ^ { 2 } } + \frac { d y } { d t } + 4 y = 0$

question 88

Multiple Choice

Solve the differential equation.
$\frac { d ^ { 2 } y } { d t ^ { 2 } } + \frac { d y } { d t } + 4 y = 0$

• A)
• \(y ( t ) = t e ^ { \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { 15 } t } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { 15 } t } { 2 } \right) \right]\)
• B)
• \(y ( t ) = t ^ { 2 } e ^ { - \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \sqrt { \frac { 15 t } { 2 } } \right) + c _ { 2 } \sin \left( \sqrt { \frac { 15 t } { 2 } } \right) \right]\)
• C)
• \(y ( t ) = e ^ { \frac { \sqrt { t } } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { t } } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { t } } { 2 } \right) \right]\)
• D)
• \(y ( t ) = c e ^ { - \frac { t } { 2 } } \cos \left( \sqrt { \frac { 15 t } { 2 } } \right)\)
• E)
• \(y ( t ) = e ^ { - \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { 15 } t } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { 15 } t } { 2 } \right) \right]\)

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Definitions:

Diminishing Utility

A principle in economics that as consumption of a good or service increases, the marginal utility derived from each additional unit decreases.

Fed

Short for the Federal Reserve, which is the central banking system of the United States, responsible for monetary policy.

Externalities

Costs or benefits that affect parties who did not choose to incur that cost or benefit, often leading to market failure if unaddressed.

Moral Hazard

A situation in which one party engages in risky behavior or lacks incentive to guard against risk because another party bears the consequences.