( a ) Write the system of equations in the form Y( t ) = ( 1/6 ) t 3 e -2t + e -2t + 3t e -tĥ Given a system of homogenous linear differential equations ( c ) Using Laplace transform, prove that the particular ( b ) Use the Laplace transform to solve the following initial-value problem ( iii ) What is the charge on the capacitor whenĤ( a ) Solve the initial value problem of Cauchy-Euler ( i ) Show that the general solution of the charge With L = 1 Henry, R= 20 ohms, C= 0.0005 farad L d 2 q / d t 2 + Rdq/dt + ( 1 / C ) q = E( t ) Hint : Kirchoff's Second Law, RI + ( 1/C ) q =E( t ) ( b ) A 250-Volt electromotive force is applied toĪn RC series circuit where the resistance isġ000 ohms and the capacitance is 5x 10 -6 farad.įind the charge q( t ) on capacitor if the ( c ) Find the explicit solution for the followingģ ( a ) The population of a town P( t ) is modelled byĪt time t.The initial population of 5000 increaseīy 20% in 5 years. Is an exact equation.Hence ,find the general solution ( e xsinx-2ysinx ) dx = -(2cosx + e xcosy)dy Where C is a constant, is the general solutionĢ( a ) Verify that the integrating factor of ( ii ) S = c R 3ġ ( a ) Find the solution for the following differential equation x y ' + 2y = x 3, y( 1 ) = 2
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