Determine the order and degree (if exists) of the following differential equations: 
\(\frac { dy }{ dx } =x+y+5\)

Determine the order and degree (if exists) of the following differential equations: 
\({ \left( \frac { { d }^{ 4 }y }{ { dx }^{ 4 } } \right) }^{ 3 }+4{ \left( \frac { dy }{ dx } \right) }^{ 7 }+6y=5cos3x\)

Determine the order and degree (if exists) of the following differential equations: 
\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3{ \left( \frac { dy }{ dx } \right) }^{ 2 }={ x }^{ 2 }log\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \)

Determine the order and degree (if exists) of the following differential equations: 
\(3\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ={ \left[ 4+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 3 }{ 2 } }\)

Determine the order and degree (if exists) of the following differential equations: 
dy + (xy − cos x)dx = 0

Show that x² + y² = r², where r is a constant, is a solution of the differential equation \(\frac { dy }{ dx } \) = -\(\frac { x }{ y } \).

Show that y = mx + \(\frac{7}{m}\), m ≠ 0 is a solution of the differential equation xy'+7\(\frac{1}{y'}\)-y = 0.

Show that y = 2(x²−1)+Ce^−x2 is a solution of the differential equation \(\frac { dy }{ dx } +2xy-4{ x }^{ 3 }=0\)

Show that y = a cos(log x) + bsin (log x), x > 0 is a solution of the differential equation x² y" + xy'+y = 0.

Solve \(\left( y+\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \right) dx-xdy=0,\ y(1)=0\)

Solve (2x + 3y)dx + (y − x)dy = 0.

Solve \({ y }^{ 2 }+{ x }^{ 2 }\frac { dy }{ dx } =xy\frac { dy }{ dx } \)

Solve \((1+{ 2e }^{ x/y })dx+2{ e }^{ x/y }\left( 1-\frac { x }{ y } \right) dy=0\)

A radioactive isotope has an initial mass 200mg, which two years later is 50mg. Find the expression for the amount of the isotope remaining at any time. What is its half-life? (half-life means the time taken for the radioactivity of a specified isotope to fall to half its original value).

In a murder investigation, a corpse was found by a detective at exactly 8 p.m. Being alert, the detective also measured the body temperature and found it to be 70^oF. Two hours later, the detective measured the body temperature again and found it to be 60^oF. If the room temperature is 50^oF, and assuming that the body temperature of the person before death was 98.6^oF, at what time did the murder occur? [log(2.43) = 0.88789; log(0.5)=-0.69315]