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Hello everyone.
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In this question we need to solve the differential equation y double prime plus 3 t y prime minus 6 y equals to 2.
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Note that y double prime means the double derivative of y with respect to t that is d square y over d t squared and y prime denotes the first order derivative of y with respect to t that is d y over t the initial condition is given that is at t equal to zero y is zero and the first order derivative that is y prime or ty over t t at t equal to zero is zero given the initial condition we need to solve this differential equation using the laplace transform.
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The laplace transform is an integral transform that converts a function of a real variable.
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Here the function is given by ft and it is in the time domain.
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The laplace transform converts ft to a function in the complex variable s and the integral transform is given by the integration from the limit 0 to infinity fd, fd, and multiplied by e -r -r -r -res to the power minus s multiplied by t t t now let us take the laplace transform on the left -hand side as well as on the right -hand side of the differential equation therefore we shall obtain the laplace transform denoted by capital l of y double prime plus the laplace transform of three multiplied by d multiplied by y prime minus the laplace transform of 6 multiplied by y and this is equal to the laplace transform of 2 in order to proceed further let us note down some of the important results of a laplace transform from the laplace transform of the derivative of a function we have the laplace transform of y prime is equal to s multiplied by the laplace transform of y minus the value of y at t equals to zero therefore if we do the laplace transform of y double prime this will be equal to s multiplied by the laplace transform of y prime minus y prime zero and on further calculation we can obtain s squared multiplied by the laplace transform of y minus s multiplied by y 0 minus y prime 0.
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Also the laplace transform of any function which is given as d multiplied by fy will be equal to negative of the derivative of the laplace transform of fy with respect to s.
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And note that the laplace transform of 1 will be equal to 1 divided by s.
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Now if we use this formula then we shall obtain for the first term the laplace transform of y double prime is equal to s squared multiplied by the laplace transform of y minus s multiplied by y 0 minus y prime 0 now in this problem it is given that y 0 and y prime 0 therefore we shall obtain the laplace transform of y double prime as s squared multiplied by the laplace transform of y and let us denote this as equation 1.
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Now we have to find the laplace transform of 3 multiplied by t multiplied by y prime so 3 is constant we can take it cite the laplace transform so it will be 3 multiplied by laplace transform of t multiplied by y prime now we use our formula and we shall get 3 multiplied by minus the derivative of the laplace transform of y prime with respect to s and and now if we do the simplification, we shall obtain minus 3 d over d s of s multiplied by the laplace transform of y minus y 0.
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Now y 0 is 0 in our problem.
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So we have minus 3 d over ds of s multiplied by the laplace transform of y.
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Now if we use the product rule of differentiation, we shall obtain the expression minus 3 multiplied by the laplace transform of y minus 3 multiplied by the laplace transform of y minus 3 multiplied by s minus 3 multiplied by s minus d over d s of the laplace transform of y similarly the laplace transform of two that is the right hand side of the differential equation will give me two divided by s so let us denote these two equations as equation two and equation so if we put equation 1, 2 and 3 in the laplace transform of the differential equation that we have written, then we shall obtain s squared multiplied by the laplace transform of y minus 3 multiplied by the laplace trans form of y minus 3 multiplied by s minus t over t.
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Of the laplace transform of y minus 6 multiplied by laplace transform of y equals to 2 divided by s or if we simplify we shall obtain dly over ds that is the first derivative of the laplace transform of y with respect to s plus 3 divided by s minus s divided by 3 multiplied by the laplace transform of y that is equal to minus 2 divided by 3 multiplied by s rest to the power 2 now we need to solve this linear first daughter differential equation and the method to solve this linear first order differential equation is to first find the integrating factor...