Solvng First Order Differential Equations : tanθdr/dθ + r = sin²θ

Please solve the following differential equation, showing all steps and substitutions. Thank you. tanθdr/dθ + r = sin²θ

Integral Equation

Solve the integral equation: y(x) = 1 + integral from 0 to x of ( y(t)*t^2 dt) (in the integral is y as a function of t times t^2)

Initial value problem

Consider the initial value problem on [1,2]: x^2*y’’ + xy’ – K^2*y = 0, y(1) =1, y’(1)=0 Find the solution y(x,K). Is it a continuous function of K? Can it be differentiated with respect to K? K is a constant

Solving a System of Differential Equations

1. Consider the initial-value problem dx/dt = A(f,u)x, x(0) =[0] [1] where A(f,u)= [ -f 1] [ 0 -u] where f and u are positive parameters. Solve it if u does not equal f

Implicit Differentiation

Suppose f=f(x), g=g(x), and I=I(x). Solve the following linear equation to get an implcit solution for I(x): fI' + (f' - g)I = 0 f>0

Find the General Solution of the Differential Equation : 2d²y/dt² + 5dy/dt + 2y = 0

Please find the general solution of the differential equation below. It needs to be converted to standard format (s² + bs + c = 0), the roots found, and then plugged into the appropriate general formula that satisfies whether b² + 4c is less than, greater than, or equal to zero. Please show all details in the solution. Thank yo ...continues

Find the general solution of the differential equation : 2d²y/dx² + 4dy/dx + 5y = 0

Please find the general solution of the differential equation below. It needs to be converted to standard format (s² + bs + c = 0), the roots found, and then plugged into the appropriate general formula that satisfies whether b² + 4c is less than, greater than, or equal to zero. Please show all details in the solution. Thank yo ...continues

Find the particular solution to the differential equation satisfying the given initial conditions : d²y/dx² + 16y = 0 y(0)=3, y´(0)=12

Please find the particular solution of the differential equation below satisfying the given initial conditions. The roots need to be found, and then plugged into the appropriate general formula that satisfies whether b² + 4c is less than, greater than, or equal to zero. After finding the general solution, solve for the values ...continues

Find the particular solution of the differential equation : 4d²y/dx² -4dy/dx + y = 0 y(0)=0, y´(0)=1

Please find the particular solution of the differential equation below satisfying the given initial conditions. The equation needs to be put in standard form (s² + bs + c = 0). The roots need to be found, and then plugged into the appropriate general formula that satisfies whether b² + 4c is less than, greater than, or equal ...continues

Find the particular solution of the differential equation : y=C(e^(-2x))sinx + C(e^(-2x))cos x y(0)= 1 y'(0)= 0

Find the particular solution of the general equation below, given the initial conditions. Please show all details in the solution, simplifying if possible. Thank you. y=C(e^(-2x))sinx + C(e^(-2x))cos x y(0)= 1 y'(0)= 0