In this page creating differential equations solution1 we are going to see solutions of some practice questions.

Form a differential equations by eliminating arbitrary constants given in brackets against each.

(i) y² = 4ax {a}

Solution:

here we have only one arbitrary constant,so we can differentiate the given equation with respect to x.

2y (dy/dx) =4a (1)

2y (dy/dx) =4a

dy/dx = 4a/2y

dy/dx = 2a/y ---- (1)

now we are going to find the value of "a" in terms of x and y from the given question.

y² = 4ax

a = y²/4x

now we are going to apply the value of a in the first equation

dy/dx = 2(y²/4x)/y

dy/dx = 2y²/4xy

y' = y/2x

2 x y' = y

y = 2 x y'

Therefore the required equation is y = 2 x y'.

(ii) y = a x² + bx + c {a,b}

Solution:

here we have two arbitrary constants,so we can differentiate the given equation with respect to x two times

y = a x² + bx + c --- (1)

differentiate with respect to x

dy/dx = a (2 x) + b (1) + 0

dy/dx = 2 a x + b

again differentiate the above equation with respect to x

d²y/dx² = 2 a (1) + 0

d²y/dx² = 2 a

y' = 2 a x + b --- (2)

y'' = 2 a --- (3)

we are going to apply the value of 2a in the second equation

y' = y'' x + b

b = y' - y'' x

a = y''/2

now we are going to apply the values of a and b in the first equation

y = a x² + b x + c

y = (y''/2)x² + (y' - y'' x)x + c

y = (y''x²/2) + y' x - y'' x² + c

y = (y''x² + 2 y' x - 2 y'' x² + 2 c)/2

2 y = 2 y' x - y'' x² + 2 c

x² y'' - 2 x y' + 2 y - 2 c = 0

Therefore the required differential equation is x² y'' - 2 x y' + 2 y - 2 c = 0

creating differential equations solution1 creating differential equations solution1

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- Properties of integrals
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- Integration-using partial fractions
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