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Vector Algebra

Question
CBSEENMA12033064
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Solve
dy over dx space equals space sin cubed straight x space cos squared straight x plus straight x space straight e to the power of straight x

Solution

The given differential equation is
              dy over dx space equals space sin cubed straight x space cos squared straight x plus straight x space straight e to the power of straight x
Separting the variables, we get,   dy space equals space left parenthesis sin cubed straight x space cos squared straight x plus straight x space straight e to the power of straight x right parenthesis space dx
therefore space space space space integral dy space equals space integral left parenthesis sin cubed straight x space cos squared straight x plus straight x space straight e to the power of straight x right parenthesis space dx therefore space space integral dy space equals space integral sin cubed straight x space cos squared straight x space dx space plus space integral straight x space straight e to the power of straight x space dx space space space space space space... space left parenthesis 1 right parenthesis
Let   straight I subscript 1 space equals space integral sin cubed straight x space space cos squared straight x space dx space equals space integral sin squared straight x space cos squared straight x. space sinx space dx
            equals space integral left parenthesis 1 minus cos squared straight x right parenthesis space cos squared straight x. space sinx space dx
Put cosx =t,     therefore space space space space sin space straight x space dx space equals space minus dt
therefore space space space straight I subscript 1 space equals space minus integral left parenthesis 1 minus straight t squared right parenthesis space straight t squared space dt space equals space minus integral left parenthesis straight t squared minus straight t to the power of 4 right parenthesis space dt space equals space minus open parentheses straight t cubed over 3 minus straight t to the power of 5 over 5 close parentheses equals 1 fifth straight t to the power of 5 minus 1 third straight t cubed
             equals space 1 fifth cos to the power of 5 straight x space minus space 1 third cos cubed straight x
Let   straight I subscript 2 space equals space integral straight x space straight e to the power of straight x space dx space equals space straight x space. straight e to the power of straight x space minus space integral 1. space straight e to the power of straight x space dx space equals space straight x space straight e to the power of straight x space minus space straight e to the power of straight x space equals space left parenthesis straight x minus 1 right parenthesis space straight e to the power of straight x
therefore space space space from space left parenthesis 1 right parenthesis comma space space integral dy space equals space 1 fifth cos to the power of 5 straight x minus 1 third cos cubed straight x plus left parenthesis straight x minus 1 right parenthesis space straight e to the power of straight x therefore space space space straight y space equals space 1 fifth cos to the power of 5 straight x space minus space 1 third cos cubed straight x space plus space left parenthesis straight x minus 1 right parenthesis space straight e to the power of straight x plus straight c space is space the space required space solution. space      

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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