First-Order Equations, Linear Equations- Problem Set

First-Order Equations, Linear Equations- Problem Set -...

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1 First-Order Equations: Linear Equations- Problem Set Solve the following linear first-order differential equations. 1. '2 2 2 t yy t e −= Comparing this equation with the general linear first-order equation, () dy pty gt dx += , we can see that, 2 pt = − , 22 t gt te = . Therefore, the integrating factor is: 2 2 p t dt dt t µµ µ ∫∫ =⇒ = = Multiplying both side of the equation by the integrating factor, we get: 3 2' 2 2 2 2 2 ' 2 2 2 2 2 2 2( ) ( ) ( ) 3 tt t t t t t t dt ey ey e t e ey t d t ey C dt −− = = = ⇒= + 3 2 3 t t ye C =+ 2. (1 ) 4 ) ty t y t ++ = + ' 41 ) 4 ) 11 t y t y y = +⇒ + = + + Comparing this equation with the general linear first-order equation, dy dx , we can see that: 2 4 1 t t = + . Therefore, the integrating factor is: ptd t 2 2 4 2ln(1 ) ln(1 ) 2 2 1 ) t dt t e e t + == = = + Multiplying both side of the equation by the integrating factor, we get: 22 ' 2 2 ' 2 2 ) 4 (1 ) 1 ((1 ) ) 1 ) ) ty t t y t t t d t + + + =+⇒ + =+⇒+ = + 1 ) tan ( ) tC + 1 tan ( ) ) y t + = + 3.
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First-Order Equations, Linear Equations- Problem Set -...

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