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Vector Algebra

Question
CBSEENMA12035779
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Prove that x2 – y2 = c(x2 + y2)2 is the general solution of the differential equation (x3 – 3xy2)dx = (y3 – 3x2y) dy, where C is a parameter.

Solution

Given equation is
(x3 – 3xy2)dx = (y3 – 3x2y) dy
rightwards double arrow dy over dx space equals space fraction numerator straight x cubed minus 3 xy squared over denominator straight y cubed minus 3 straight x squared straight y end fraction
which is a homogeneous equation.
Therefore substituting y = vx
We have,
dy over dx space equals space straight v space plus straight x. dv over dx straight v plus straight x. dv over dx space equals space fraction numerator straight x cubed minus 3 straight x. straight v squared straight x squared over denominator straight v cubed straight x cubed minus 3 straight x squared. vx end fraction straight v plus straight x. dv over dx space equals space fraction numerator 1 minus 3 straight v squared over denominator straight v cubed minus 3 straight v end fraction straight x. dv over dx space equals space fraction numerator begin display style 1 minus 3 straight v squared end style over denominator begin display style straight v cubed minus 3 straight v end style end fraction space minus straight V straight x. dv over dx space equals space fraction numerator begin display style 1 minus 3 straight v squared space minus straight v to the power of 4 space plus 3 straight v squared end style over denominator straight v cubed minus 3 straight v end fraction fraction numerator straight v cubed minus 3 straight v over denominator 1 minus straight v to the power of 4 end fraction. dv space equals 1 over straight x. dx
Integrating the equation both sides, 
log space straight x space equals space integral fraction numerator straight v cubed minus 3 straight v over denominator 1 minus straight v to the power of 4 end fraction dv rightwards double arrow space log space straight C apostrophe space plus log space straight x space equals space integral fraction numerator straight v cubed over denominator 1 minus straight v to the power of 4 end fraction. dv minus 3. integral fraction numerator straight v over denominator 1 minus straight v to the power of 4 end fraction. dv log space straight C apostrophe straight x space equals space straight I subscript 1 minus 3. straight I subscript 2.......... space left parenthesis 1 right parenthesis space where space straight C space is space an space integration space constant straight I subscript 1 space equals space integral fraction numerator straight v cubed over denominator 1 minus straight v to the power of 4 end fraction. dv let space 1 minus straight v to the power of 4 space equals space straight t rightwards double arrow space minus 4 straight v cubed. dv space equals space dt straight I subscript 1 space equals space integral 1 over straight t. open parentheses fraction numerator dt over denominator negative 4 end fraction close parentheses space equals space minus 1 fourth. log space straight t straight I subscript 1 space equals space minus 1 fourth space lof space left parenthesis 1 minus straight v to the power of 4 right parenthesis space.... left parenthesis 2 right parenthesis straight I subscript 2 space equals space integral fraction numerator straight v over denominator 1 minus straight v to the power of 4 end fraction. dv let space straight v squared space equals space straight t rightwards double arrow space 2 straight v. dv space equals space dt straight I subscript 2 space equals space integral fraction numerator 1 over denominator 1 minus straight t squared end fraction. open parentheses dt over 2 close parentheses space equals space 1 fourth. log fraction numerator 1 plus straight t over denominator 1 minus straight t end fraction straight I subscript 2 space equals space 1 fourth. log space open parentheses fraction numerator 1 plus straight v squared over denominator 1 minus straight v squared end fraction close parentheses
thus substituting the values of I1 and I2, 
log space straight C apostrophe straight x space equals space minus 1 fourth. log space left parenthesis 1 minus straight v to the power of 4 right parenthesis minus 3 over 4. log space open parentheses fraction numerator 1 plus straight v squared over denominator 1 minus straight v squared end fraction close parentheses log space straight C apostrophe straight x space equals space minus space 1 fourth. space open square brackets log space open parentheses 1 minus straight v to the power of 4 close parentheses minus 3 over 4. log open parentheses fraction numerator 1 plus straight v squared over denominator 1 minus straight v squared end fraction close parentheses cubed close square brackets log space straight C apostrophe straight x space equals space minus 1 fourth. open square brackets log space open parentheses 1 minus straight v to the power of 4 close parentheses. open parentheses fraction numerator 1 plus straight v squared over denominator 1 minus straight v squared end fraction close parentheses cubed close square brackets log space straight C apostrophe straight x space equals space minus 1 fourth. space log space open square brackets left parenthesis 1 minus straight v squared right parenthesis left parenthesis 1 plus straight v squared right parenthesis fraction numerator left parenthesis 1 plus straight v squared right parenthesis cubed over denominator left parenthesis 1 minus straight v squared right parenthesis cubed end fraction close square brackets log space straight C apostrophe straight x space equals space 1 fourth. log space open square brackets fraction numerator left parenthesis 1 plus straight v squared right parenthesis to the power of 4 over denominator left parenthesis negative straight v squared right parenthesis squared end fraction close square brackets space equals space log space open square brackets fraction numerator left parenthesis 1 plus straight v squared right parenthesis to the power of 4 over denominator left parenthesis 1 minus straight v squared right parenthesis squared end fraction close square brackets to the power of 1 divided by 4 end exponent space equals space log space open square brackets fraction numerator left parenthesis 1 minus straight v squared right parenthesis to the power of 1 divided by 2 end exponent over denominator left parenthesis 1 plus straight v squared right parenthesis end fraction close square brackets rightwards double arrow straight C apostrophe straight x space equals space fraction numerator left parenthesis 1 minus straight v squared right parenthesis to the power of 1 divided by 2 end exponent over denominator left parenthesis 1 plus straight v squared right parenthesis end fraction straight C apostrophe straight x space equals space fraction numerator open parentheses 1 minus begin display style straight y squared over straight x squared end style close parentheses to the power of 1 divided by 2 end exponent over denominator 1 plus begin display style straight y squared over straight x squared end style end fraction straight C apostrophe straight x space equals space fraction numerator begin display style fraction numerator left parenthesis straight x squared minus straight y squared right parenthesis to the power of 1 divided by 2 end exponent over denominator straight x end fraction end style over denominator left parenthesis straight x squared plus straight y squared right parenthesis divided by straight x squared end fraction space equals space fraction numerator left parenthesis straight x squared minus straight y squared right parenthesis to the power of 1 divided by 2 end exponent over denominator left parenthesis straight x squared plus straight y squared right parenthesis end fraction straight C apostrophe left parenthesis straight x squared plus straight y squared right parenthesis space equals space left parenthesis straight x squared minus straight y squared right parenthesis squaring space both space sides space straight C apostrophe left parenthesis straight x squared plus straight y squared right parenthesis squared space equals space left parenthesis straight x squared minus straight y squared right parenthesis

Some More Questions From Vector Algebra Chapter

Determine order and degree (if defined) of differential equations:
y' + y = ex

Determine order and degree (if defined) of differential equations:
y'' + (y')2 + 2 y = 0

Determine order and degree (if defined) of differential equations:
y″ + 2y′ + sin y = 0

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