Crazy diffy! 2

Calculus Level 5

$\frac { { d }^{ 2 }x }{ { d }y^{ 2 } } +2\frac { dx }{ dy } +5x=\sin { \left( y \right) }$

Above shows a differential equation. You are given that if $y = 0$ , then $x = 1$ and $\dfrac{dx}{dy} = 2$ . Let $A$ denote the value of $x$ when $y=30$ . Find $\lfloor 1000A \rfloor$ .

Take y in degrees.

The answer is 13.

1 solution

Aditya Kumar
Feb 11, 2016

Taking the auxiliary equation of the equation, we get:

$\overline { x } \left( p \right) \left( { p }^{ 2 }+2p+5 \right) =p+4+\mathcal{ L }\left\{ \sin { y } \right\}$

$\overline { x } \left( p \right) \left( { p }^{ 2 }+2p+5 \right) =p+4+\frac { 1 }{ { p }^{ 2 }+1 }$

$\overline { x } \left( p \right) =\frac { p+4 }{ { p }^{ 2 }+2p+5 } +\frac { 1 }{ \left( { p }^{ 2 }+1 \right) \left( { p }^{ 2 }+2p+5 \right) }$

$\overline { x } \left( p \right) =\frac { \frac { 11 }{ 10 } p+4 }{ { p }^{ 2 }+2p+5 } +\frac { \frac { -1 }{ 10 } p+\frac { 1 }{ 5 } }{ \left( { p }^{ 2 }+1 \right) }$

$\overline { x } \left( p \right) =\frac { 11 }{ 10 } \frac { p+1 }{ { \left( p+1 \right) }^{ 2 }+{ 2 }^{ 2 } } +\frac { 29 }{ 20 } \frac { 2 }{ { \left( p+1 \right) }^{ 2 }+{ 2 }^{ 2 } } \frac { \frac { -1 }{ 10 } p+\frac { 1 }{ 5 } }{ \left( { p }^{ 2 }+1 \right) }$

Therefore,

$\boxed{x\left( y \right) ={ e }^{ -y }\left( \frac { 11 }{ 10 } \cos { \left( 2y \right) } +\frac { 29 }{ 20 } \sin { \left( 2y \right) } \right) -\frac { 1 }{ 10 } \cos { \left( y \right) } +\frac { 1 }{ 5 } \sin { \left( y \right) }}$

i got the same general form but did a mistake in finding the coefficients

nice problem :)

Hamza A - 5 years, 4 months ago

Thanks. This problem is prone to mistakes. The coefficients are not at all good looking.

Aditya Kumar - 5 years, 4 months ago

Crazy diffy! 2

The answer is 13.

1 solution

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