{"id":43184,"date":"2024-04-15T21:25:50","date_gmt":"2024-04-15T21:25:50","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=43184"},"modified":"2024-04-15T21:25:50","modified_gmt":"2024-04-15T21:25:50","slug":"input-xt-and-output-yt-of-an-lti-system-are-related-by-the-differential-equation-yt-yt-6yt-xt-if-the-system-is-neither-causal-nor-stable-the-impulse-response-ht-of-the-system","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/input-xt-and-output-yt-of-an-lti-system-are-related-by-the-differential-equation-yt-yt-6yt-xt-if-the-system-is-neither-causal-nor-stable-the-impulse-response-ht-of-the-system\/","title":{"rendered":"Input x(t) and output y(t) of an LTI system are related by the differential equation y&#8221;(t) &#8211; y'(t) &#8211; 6y(t) = x(t). If the system is neither causal nor stable, the impulse response h(t) of the system is"},"content":{"rendered":"<p>\r\n    <!-- Check if it's an AMP page -->\r\n            <!-- Non-AMP version -->\r\n        <div class=\"mcq-container\" data-quiz-id=\"quizState_6a971dd03eeb6\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    $${1 over 5}{e^{3t}}uleft( { - t} \right) + {1 over 5}{e^{ - 2t}}uleft( { - t} \right)$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    $$ - {1 over 5}{e^{3t}}uleft( { - t} \right) + {1 over 5}{e^{ - 2t}}uleft( { - t} \right)$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"true\">\r\n                    $${1 over 5}{e^{3t}}uleft( { - t} \right) - {1 over 5}{e^{ - 2t}}uleft( t \right)$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    $$ - {1 over 5}{e^{3t}}uleft( { - t} \right) - {1 over 5}{e^{ - 2t}}uleft( t \right)$$                <\/div>\r\n                            \r\n            <!-- Feedback messages for non-AMP -->\r\n            <div class=\"feedback\" data-feedback=\"wrong\">Answer is Right!<\/div>\r\n            <div class=\"feedback\" data-feedback=\"right\">Answer is Wrong!<\/div>\r\n        <\/div>\r\n\r\n        <script>\r\n        document.addEventListener('DOMContentLoaded', function () {\r\n            var containers = document.querySelectorAll('.mcq-container');\r\n\r\n            containers.forEach(function(container) {\r\n                var options = container.querySelectorAll('.option');\r\n                var feedbackSelect = container.querySelector('[data-feedback=\"select\"]');\r\n                var feedbackWrong = container.querySelector('[data-feedback=\"wrong\"]');\r\n                var feedbackRight = container.querySelector('[data-feedback=\"right\"]');\r\n\r\n                options.forEach(function(option) {\r\n                    option.addEventListener('click', function() {\r\n                        var selectedOption = option.getAttribute('data-option-key');\r\n                        var isCorrect = option.getAttribute('data-is-correct') === 'true';\r\n\r\n                        \/\/ Remove previous selections\r\n                        options.forEach(function(opt) {\r\n                            opt.classList.remove('correct', 'incorrect');\r\n                        });\r\n\r\n                        \/\/ Add the correct\/incorrect class\r\n                        if (isCorrect) {\r\n                            option.classList.add('correct');\r\n                            feedbackRight.hidden = false;\r\n                            feedbackWrong.hidden = true;\r\n                        } else {\r\n                            option.classList.add('incorrect');\r\n                            feedbackRight.hidden = true;\r\n                            feedbackWrong.hidden = false;\r\n                        }\r\n\r\n                        \/\/ Hide select feedback\r\n                        feedbackSelect.hidden = true;\r\n                    });\r\n                });\r\n            });\r\n        });\r\n        <\/script>\r\n    \r\n    <!--more--><\/p>\n<p>The correct answer is $\\boxed{{1 \\over 5}{e^{3t}}u\\left( { &#8211; t} \\right) + {1 \\over 5}{e^{ &#8211; 2t}}u\\left( { &#8211; t} \\right)}$.<\/p>\n<p>The impulse response of an LTI system is the response of the system to a unit impulse input. The differential equation $y&#8221;(t) &#8211; y'(t) &#8211; 6y(t) = x(t)$ can be solved using the method of undetermined coefficients. The general solution to this equation <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div> is $y(t) = Ae^{3t} + Be^{-2t}$, where $A$ and $B$ are constants to be determined.<\/p>\n<p>To determine $A$ and $B$, we need to know the initial conditions of the system. In this case, we are told that the system is neither causal nor stable. This means that the system does not have a zero initial condition, i.e., $y(0) \\neq 0$.<\/p>\n<p>The initial conditions of the system can be found by taking the derivatives of the general solution and evaluating them at $t = 0$. The first derivative of $y(t)$ is $y'(t) = 3Ae^{3t} &#8211; 2Be^{-2t}$. The second derivative of $y(t)$ is $y&#8221;(t) = 9Ae^{3t} + 4Be^{-2t}$.<\/p>\n<p>Evaluating these derivatives at $t = 0$, we get $y'(0) = 3A$ and $y&#8221;(0) = 9A$. We are also given that $x(t) = u(t)$, where $u(t)$ is the unit step function. The unit step function is defined as $u(t) = 0$ for $t &lt; 0$ and $u(t) = 1$ for $t \\geq 0$.<\/p>\n<p>The response of the system to the unit step input is $y(t) = Ae^{3t} + Be^{-2t}$. To find $A$ and $B$, we need to substitute this expression into the differential equation $y&#8221;(t) &#8211; y'(t) &#8211; 6y(t) = x(t)$. This gives us the following equation:<\/p>\n<p>$$9Ae^{3t} &#8211; 2Be^{-2t} &#8211; 3Ae^{3t} + 2Be^{-2t} &#8211; 6(Ae^{3t} + Be^{-2t}) = 1$$<\/p>\n<p>Simplifying this equation, we get $-3A &#8211; 4B = 1$. Solving for $A$ and $B$, we get $A = {1 \\over 5}$ and $B = {1 \\over 5}$.<\/p>\n<p>Therefore, the impulse response of the system is $h(t) = {1 \\over 5}{e^{3t}}u\\left( { &#8211; t} \\right) + {1 \\over 5}{e^{ &#8211; 2t}}u\\left( { &#8211; t} \\right)$.<\/p>\n<p>The other options are incorrect <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> because they do not satisfy the differential equation $y&#8221;(t) &#8211; y'(t) &#8211; 6y(t) = x(t)$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Join Our Telegram Channel Subscribe on YouTube<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[959],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Input x(t) and output y(t) of an LTI system are related by the differential equation y&quot;(t) - y&#039;(t) - 6y(t) = x(t). 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