Take Laplace transform on both the sides of the given differential equation.


The necessary formulas of Laplace transform of the derivatives of a function are applied.

Most commonly used are:


Substituting all the necessary Laplace transform, the expression of is obtained.


Once  is found, then  can be obtained by taking Inverse Laplace transform.



Steps to find the solution of ordinary differential equations using Laplace Transforms


  1. Solve the differential equation using Laplace transform (d^2 x)/(dt^2 )+3 dx/dt+2x=e^(-t) given the initial conditions x(0)=0 & x'(0)=1
  2. Solve the equations (d^2 y)/ (dt^2) +5 dy/dt+6y=5e^2t given that y (0) =2, dy/dt (0) =1
  3. Solve the equation (d^2 y)/ (dt^2) +2 dy/dt-3y=sin t under the conditions y(0) = dy/dt (0) =0
  4. Solve the equation (d^2 y)/ (dt^2) – dy/dt = 0 under the conditions y(0) = y’(0) =3
  5. Solve the equation (d^2 y)/ (dt^2) + 3 dy/dt + 2 y(t) = 0 under the conditions y(0) = 1 & y’(0) =0
  6. Solve the initial problem y”+2y’+2y = 5 sin t when y(0)=y'(0)=0


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