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how to solve this problem

Asked by ganilkumar726 | 09 Mar, 2022, 09:53: PM

Expert Answer

To find the solution of the differential equation: y" - 10y' + 9y = 5t, y(0) = -1, y'(0) = 2

Let space straight m space be space the space root space of space thie space auxiliary space equation space straight s. straight t. straight m squared minus 10 straight m plus 9 equals 0 rightwards double arrow open parentheses straight m minus 9 close parentheses open parentheses straight m minus 1 close parentheses equals 0 rightwards double arrow straight m equals 9 space and space straight m space equals space 1 The space particular space solution space will space be space straight y subscript straight c equals straight c subscript 1 straight e to the power of straight t plus straight c subscript 2 straight e to the power of 9 straight t end exponent Solution space will space be space of space the space form colon space straight y subscript straight p equals straight a subscript 1 plus straight a subscript 2 straight t rightwards double arrow straight y subscript straight p apostrophe equals straight a subscript 2 rightwards double arrow straight y subscript straight p " equals 0 Putting space the space values space in space the space DE comma space we space get minus 10 straight a subscript 2 plus 9 open parentheses straight a subscript 1 plus straight a subscript 2 straight t close parentheses equals 5 straight t rightwards double arrow 9 straight a subscript 1 minus 10 straight a subscript 2 plus 9 straight a subscript 2 straight t equals 5 straight t rightwards double arrow 9 straight a subscript 1 minus 10 straight a subscript 2 equals 0 space and space 9 straight a subscript 2 straight t equals 5 straight t Solving space them comma space we space get straight a subscript 1 equals 50 over 81 space space and space space straight a subscript 2 equals 5 over 9 Substituting space straight a subscript 1 space and space straight a subscript 2 space in space straight y subscript straight p comma space we space get straight y subscript straight p equals fraction numerator 5 straight t over denominator 9 end fraction plus 50 over 81 So comma space the space general space solution space is colon space straight y equals straight y subscript straight c plus straight y subscript straight p equals straight c subscript 1 straight e to the power of straight t plus straight c subscript 2 straight e to the power of 9 straight t end exponent plus fraction numerator 5 straight t over denominator 9 end fraction plus 50 over 81 rightwards double arrow dy over dt equals straight c subscript 1 straight e to the power of straight t plus 9 straight c subscript 2 straight e to the power of 9 straight t end exponent plus 5 over 9 Using space straight y left parenthesis 0 right parenthesis equals negative 1 space in space the space general space solution space " straight y " comma space we space have straight c subscript 1 plus straight c subscript 2 plus 50 over 81 equals negative 1 Using space straight y apostrophe left parenthesis 0 right parenthesis equals 2 space in space dy over dt comma space we space get straight c subscript 1 plus 9 straight c subscript 2 plus 5 over 9 equals 2 SOlving space the space two space equations comma space we space get straight c subscript 1 equals negative 2 space and space straight c subscript 2 equals 31 over 38 Thus comma space the space solution space will space be straight y equals 1 over 81 open parentheses 31 straight e to the power of 9 straight t end exponent minus 162 straight e to the power of straight t plus 45 straight t plus 50 close parentheses

Answered by Renu Varma | 10 Mar, 2022, 11:18: AM

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