Find value of m so that the function y = e^mx is a solution of the given differential equation, y''− 5y' + 6y = 0

Show that y = a cos bx is a solution of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ b }^{ 2 }y=0\).

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

Find the differential equation of the family of parabolas y² = 4ax, where a is an arbitrary constant.

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.

Solve:\(\frac { dy }{ dx } \) = (3x+y+4)².

Solve the following differential equations or show that the solution of 
\(\\ \\ \\ \frac { dy }{ dx } =\sqrt { \frac { 1-{ y }^{ 2 } }{ 1-{ x }^{ 2 } } } \)

Solve the Linear differential equation:
\(\frac { dy }{ dx } =\frac { { sin }^{ 2 }x }{ 1+{ x }^{ 3 } } -\frac { { 3x }^{ 2 } }{ 1+{ x }^{ 3 } } y\)