Explanation Approach: a) An equation is homogeneous when the right hand side of the equation can be expressed as a function of the ratio of y x . i.e. d y d x = f ( x , y ) b) An equation is a Bernoulli equation when an equation can be represented in the form of d y d x + P ( x ) y = Q ( x ) y n . Here, P ( x ) and Q ( x ) are continuous on an interval ( a , b ) and is n the real number. c) An equation is the equation with linear coefficients when it can be represented in the form of ( a 1 x + b 1 y + c 1 ) d x + ( a 2 x + b 2 y + c 2 ) d y = 0 . Here, a i ' , b i ' , and c i ' are constants. d) An equation can be represented in the form of y ' = G ( a x + b y ) when the right hand side of the equation y ' can be represented as a function of the combination of a x + b y . Here, a and b are constants. Calculation: Check for the homogeneous equation. Consider the given equation. 2 t x d x + ( t 2 − x 2 ) d t = 0 Represent the right hand side of the equation 2 t x d x + ( t 2 − x 2 ) d t = 0 as a function of the ratio. 2 t x d x = − ( t 2 − x 2 ) d t d x d t = − ( t 2 − x 2 ) 2 t x d x d t = ( x 2 − t 2 ) 2 t x = 1 2 ( x 2 x t − t 2 x t ) Further simplify the equation. d x d t = 1 2 ( x t − t x ) = 1 2 ( x t − 1 x t ) From the above equation, it is clear that the right hand side of the equation can be represented as a function of the ratio of y x . Therefore, the equation 2 t x d x + ( t 2 − x 2 ) d t = 0 is a homogeneous equation. Check for the Bernoulli equation

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Chapter 2.6, Problem 1E

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The equation 2txdx+(t2−x2)dt=0 is homogeneous, Bernoulli, linear coefficient or of the form y′=G(ax+by).

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Chapter 2.5, Problem 20E

Chapter 2.6, Problem 2E

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To classify: The equation 2 t x d x + ( t 2 − x 2 ) d t = 0 is homogeneous, Bernoulli, linear coefficient or of the form y ′ = G ( a x + b y ) .

To classify: The equation 2txdx+(t2−x2)dt=0 is homogeneous, Bernoulli, linear coefficient or of the form y′=G(ax+by).

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Chapter 2 Solutions

To classify:

The equation 2txdx+(t2−x2)dt=0 is homogeneous, Bernoulli, linear coefficient or of the form y′=G(ax+by).