hello need this please

exercise 1

Find the partial derivative of the dependent variable with respect to each of the independent variables. z= (7x2 + xy3 ) * dzSolve the given differential equation. 9ydx - 9xdy + x dx = 8dx The solution is (Type an equation.)Find the particular solution to the given differential equation that satisfies the given conditions. 4 2 X +y dy = 9(xdx + ydy); x =0 when y = 1 The particular solution is (Type an equation.)Solve the given differential equation. The form of yo is given. Day - 6Dy + By = -24 (Let yp = A.) Choose the correct solution below. c, and co are arbitrary constants. O A. y=Ce + Cge -24 O B. y= C12x + cze4x - 3 OC. y= Cyex + cze 4 x OD. y=Ge* + cge 4% - 32xFind the general solution of the given equation. y" + By' + 15y =0 Choose the correct solution below. c, and c2 are arbitrary constants. OA. y=Ge +coxex O B. y=Cye 3* + Czek O C. y= c, cos ( - 3x) + cq sin ( - 5x) OD. y=Ge" - 5XFor a given circuit, L =2 mH, R=0, C=90 nF, and E =0. Find the equation relating the charge and the time if q = 10" C and i =0 when t= 0. Choose the correct equation below. O A. q= 100,000 cos 0.0745t O B. q= 100,000 cos 74,536t O C. q= cos 74,536t O D. q = 100,000 sin 74,536tThe volume V of a right circular cylinder is given by V= ar h, where r is the radius and h is the height. If h is held fixed at h = 10 inches, find the rate of change of V with respect to r when r= 10 inches. The rate is in . (Type an exact answer in terms of It.)Integrate the function. 3 12 cos x sin x dx 12 Cos x sin x dx =Integrate the function. 1 (2 + 2 x ) 5 e dx 1 ( 2 + 2x ) " edxIntegrate the given function. 15e 5x dx e 5%% + 2 5x dx e 5%% + 2For the circuit shown to the right, after the switch is closed, the transient current i (in A) is given by i= 2.0 (1 + e" "). Find the first three nonzero terms of the Maclaurin expansion WW of this function. 20 0 20 92 - 2.0 UF 80 V The first nonzero term of the Maclaurin expansion is. (Type an expression using t as the variable.)3x The current i in a certain electric circuit is i = 14 sin xt. Write the first three terms of the Taylor series of this function about t= 7 Choose the correct answer below. 3x2 32 2 O A. 14 sin 2 + 14 cos 2 t - 2 - 14 sin - 2 t - 2 2 3x2 3x O B. 14 sin - 14x- sin t - 2 + 14x COS 2 1 - 2 2 2 2 3x2 3x2 3x O C. 14 sin 2 + 14x COS 2 1 - 2 - 7x- sin 2 t - 2 3x2 3x 2 - 7 sin t - + .. O D. 14 sin 2 + 14 cos 2 1 - 2 2 2Find at least three nonzero terms (including a, and at least two cosine terms and two sine terms if they are not all zero) of the Fourier series for the given function, and sketch at least three periods of the function. f( x ) =. 10 - MEX