Solving Differential Equations with the Method of Laplace Transforms Part II: Discontinuous Functions Differential Equations Lab by Karen Donnelly karend@saintjoe.edu Saint Joseph's College All Rights Reserved This worksheet demonstrates applies the method Laplace Transforms to initial value problems where the differential equation involves a function with jump discontinuities on the right hand side of the differential equation. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEoc3R1ZGVudEYnRjZGOUZARitGKw== LUkmYWxpYXNHJSpwcm90ZWN0ZWRHNiMvLUkiWUc2IjYjSSJzR0YpLUkobGFwbGFjZUdGKTYlLUkieUdGKTYjSSJ0R0YpRjJGKw== Pkk7X0VudlVzZUhlYXZpc2lkZUFzVW5pdFN0ZXBHNiJJJXRydWVHJSpwcm90ZWN0ZWRH LUkmYWxpYXNHJSpwcm90ZWN0ZWRHNiMvSSJ1RzYiSSpIZWF2aXNpZGVHNiRGJEkoX3N5c2xpYkdGKA==

Certain applications (as in the two examples that follow) involve differential equations where the function on the right hand side has a jump discontinuity in the dependent variable. Functions with jump discontinuities can be represented by using the unit step function u(t) -- also called the Heaviside function. We used the Maple alias command above to use the name u for this function. 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 LUklcGxvdEc2IjYqLUkidUdGJDYjSSJ0R0YkL0YpOyEiJCIiJDshIiIiIiMvSSZjb2xvckdGJEklYmx1ZUdGJC9JKGRpc2NvbnRHRiRJJXRydWVHJSpwcm90ZWN0ZWRHL0klYXhlc0dGJEkmZnJhbWVHRiQvSShzY2FsaW5nR0YkSSxjb25zdHJhaW5lZEdGJC9JKnRoaWNrbmVzc0dGJEYw The following represents the use of the unit step function to represent the function f that is 3 for t < 2; then jumps by -2 (down to 1) at t = 2, and finally jumps by t-1 ( from 1 to t ) at t = 5: PkkiZkc2ImYqNiNJInRHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwoIiIkIiIiLUkidUdGJDYjLCY5JEYtISIjRi1GMyomLCZGMkYtISIiRi1GLS1GLzYjLCZGMkYtISImRi1GLUYtRiRGJEYk LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYpLUkiZkdGJzYjSSJ0R0YnL0YsOyIiISIjNy9GKkYuL0koZGlzY29udEc2JUYlRiZGJkkldHJ1ZUdGJS9JJmNvbG9yR0YnSSVibHVlR0YnL0koc2NhbGluZ0dGJ0ksY29uc3RyYWluZWRHRicvSSp0aGlja25lc3NHRiciIiM= We can double check our definition of the function f for correctness by converting it to a piecewise definition: LUkoY29udmVydEclKnByb3RlY3RlZEc2JC1JImZHNiI2I0kidEdGKEkqcGllY2V3aXNlR0Yk In order to solve differential equations whose right hand side involve discontinuities using the method of Laplace transforms we will need to know how to compute the Laplace Transform of functions such as the one above. The useful transform formulas are (using F(s) is the Laplace transform of f(t)): Function Laplace Transform 1. 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 3. 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 Formula (1) is best for computing the inverse Laplace transform and the second formula (2) is best for computing the Laplace Transform (even though they are equivalent to each other). They are easily verified using the definition of Laplace Transforms. Two Examples 1. Compute the Laplace transform of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtSSVtc3VwR0YkNiUtRiw2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GPVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRkMvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRk4vJSlzdHJldGNoeUdGTi8lKnN5bW1ldHJpY0dGTi8lKGxhcmdlb3BHRk4vJS5tb3ZhYmxlbGltaXRzR0ZOLyUnYWNjZW50R0ZOLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGZ24tRiM2Jy1GLDYlUSJ1RidGOUY8LUZJNi1RMCZBcHBseUZ1bmN0aW9uO0YnRkNGTEZPRlFGU0ZVRldGWUZlbkZobi1JKG1mZW5jZWRHRiQ2JC1GIzYnRistRiM2J0Y2LUZJNi1RKCZtaW51cztGJ0ZDRkxGT0ZRRlNGVUZXRlkvRmZuUSwwLjIyMjIyMjJlbUYnL0ZpbkZdcC1GQDYkUSIxRidGQy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQ0YrRmJwRkNGQ0ZicEZDRitGYnBGQ0YrRmJwRkNGK0ZicEZD . Apply Formula (2) with g(t) = t^2 and a = 1. You should get the same answer as Maple does below. Note that you must compute the Laplace transform 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 = 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. QyQtSShsYXBsYWNlRzYiNiUqJkkidEdGJSIiIy1JInVHRiU2IywmRigiIiIhIiJGLkYuRihJInNHRiVGLg==LUknZXhwYW5kRyUqcHJvdGVjdGVkRzYjSSIlRzYi 2. Compute the inverse Laplace transform of ((e)^((-2 s)))/(s^2). Apply Formula 1, with a = 2 and F(s) = 1/s^2. You should get the same answer as Maple below. LUkraW52bGFwbGFjZUc2IjYlKiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IywkSSJzR0YkISIjIiIiRi5GL0YuSSJ0R0Yk

Consider the differential equation: 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, where g(t) = 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 We use the unit step function to represent the right-hand side of the differential equation, since it has jump discontinuities: 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 -- we plot this below just to check that we have defined it correctly. PkkiZ0c2ImYqNiNJInRHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwoLUkidUdGJDYjOSQiIiItRi02IywmRi9GMCEiIkYwISIjLUYtNiMsJkYvRjBGNUYwRjBGJEYkRiQ= LUkoY29udmVydEclKnByb3RlY3RlZEc2JC1JImdHNiI2I0kidEdGKEkqcGllY2V3aXNlR0Yk LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYpLUkiZ0dGJzYjSSJ0R0YnL0YsOyIiISIiJS9GKjshIiMiIiMvSSZjb2xvckdGJ0klYmx1ZUdGJy9JKnRoaWNrbmVzc0dGJ0Y0L0koZGlzY29udEc2JUYlRiZGJkkldHJ1ZUdGJS9JJWF4ZXNHRidJJmJveGVkR0Yn We now proceed as before to define and solve the differential equation using the method of Lapalace transforms. Step 0. Define the differential equation to be solved (ODE) along with initial conditions (IC). PkkkT0RFRzYiLywmLUklZGlmZkclKnByb3RlY3RlZEc2JS1JInlHRiQ2I0kidEdGJEYuRi4iIiJGKyIiJSwoLUkidUdGJEYtRi8tRjM2IywmRi5GLyEiIkYvISIjLUYzNiMsJkYuRi9GOEYvRi8= PkkjSUNHNiI2JC8tSSJ5R0YkNiMiIiFGKi8tLUkiREdJKF9zeXNsaWJHRiQ2I0YoRilGKg== Asking Maple to solve using the method of Lapace transforms -- so we know what answer we SHOULD get for our final answer. PkklU09MTkc2Ii1JJ2Rzb2x2ZUdGJDYlPCRJI0lDR0YkSSRPREVHRiQtSSJ5R0YkNiNJInRHRiQvSSdtZXRob2RHRiRJKGxhcGxhY2VHRiQ= This MAY be the same our by hand solution-- see if it can be simplified by combining terms using trigonometric identities: LUkoY29tYmluZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSVTT0xOR0YnSSV0cmlnR0Yk Now we proceed with a step-by-step solution as before. Step 1. Apply the Laplace transform to the entire equation. Note that laplace(y(t),t,s) is what we have denotd by Y(s) in our "by-hand solutions" -- and we have aliased above to be Y(s). PkkoT0RFX2xhcEc2Ii1JKGxhcGxhY2VHRiQ2JUkkT0RFR0YkSSJ0R0YkSSJzR0Yk Step 2: Substitute in the initial conditions: PkkoT0RFX2xhcEc2Ii1JJXN1YnNHJSpwcm90ZWN0ZWRHNiRJI0lDR0YkRiM= Step 3: Next, it is necessary to explicitly solve for the Laplace transform of the solution. PkkpU09MTl9sYXBHNiItSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkSShPREVfbGFwR0YkPCMtSSJZR0YkNiNJInNHRiQ= Expand the solution to help visualize how to convert the inverse Laplace transform: PkkpU09MTl9sYXBHNiItSSdleHBhbmRHJSpwcm90ZWN0ZWRHNiNGIw== Now we have our solution "Y(s)" -- Compute the inverse Laplace transform using invlaplace. PkklU09MTkc2Ii1JK2ludmxhcGxhY2VHRiQ2JUkpU09MTl9sYXBHRiRJInNHRiRJInRHRiQ= Use the combine function to see if this solution looks like what we might obtain by hand: LUkoY29tYmluZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSVTT0xOR0YnSSV0cmlnR0Yk Note: In a by hand solution, the above step requires first a partial fraction expansion of 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 Maple's convert(...,parfrac) can assist us. We would use this to compute the inverse Laplace transform of the three terms in 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 LUkoY29udmVydEclKnByb3RlY3RlZEc2JComSSJzRzYiISIiLCYqJEYnIiIjIiIiIiIlRi1GKUkocGFyZnJhY0dGKA== We want to plot our solution -- Noting the form of SOLN as a set of one element, we extract the right hand side of the first (and only) element of SOLN as our expression to plot. PkkieEc2Ii1JKHVuYXBwbHlHRiQ2JC1JJHJoc0clKnByb3RlY3RlZEc2IyZJJVNPTE5HRiQ2IyIiIkkidEdGJA== The plot below shows the graph of the current f(t) at time t. LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYmLUkieEdGJzYjSSJ0R0YnL0YsOyIiISIiJy9JJmNvbG9yR0YnSSZncmVlbkdGJy9JKnRoaWNrbmVzc0dGJyIiIw==

In this problem we have a tank containing a brine solution. The tank initially contains 300 kg of salt. Initially 10 liters of solution with 4 kg of salt per liter of solution per minute is entering the tank through valve A. We close valve A and open value B at time t = 10 minutes. The solution then entering at a rate of 10 liters of solution with 2 kg of salt per liter. Hence the rate of input of salt changes from 6*4 kg/min to 6* 2 kg/min at time t = 10. The solution is kept well stirred in the tank and leaves at a rate of 10 Liters per minute. This leads to a differential equation whose right hand side has a jump discontinuity. 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 The differential equation modeling this problem is given by: 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 = 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 kilograms. Define the differential equation to be solved (ODE) along with initial conditions (IC) PkkkT0RFRzYiLywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInlHRiQ2I0kidEdGJEYuIiIiLUkiKkdGKTYkLCRGKyIiJCNGLyIkKyZGLywmLUkidUdGJEYtIiNDLUY5NiMsJkYuRi8hIzVGLyEjNw== PkkjSUNHNiIvLUkieUdGJDYjIiIhIiQrJA== Maple's solution with dsolve: LUknZHNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiU8JEkjSUNHRidJJE9ERUdGJy1JInlHRic2I0kidEdGJy9JJ21ldGhvZEdGJ0kobGFwbGFjZUdGJw== Now our step by step solution: Step 1. Apply the Laplace transform to the entire equation. PkkoT0RFX2xhcEc2Ii1JKGxhcGxhY2VHRiQ2JUkkT0RFR0YkSSJ0R0YkSSJzR0Yk Step 2: Substitute in the initial conditions: PkkoT0RFX2xhcEc2Ii1JJXN1YnNHJSpwcm90ZWN0ZWRHNiRJI0lDR0YkRiM= Step 3: Next, it is necessary to explicitly solve for the Laplace transform of the solution: PkkpU09MTl9sYXBHNiItSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkSShPREVfbGFwR0YkPCMtSSJZR0YkNiNJInNHRiQ= To help us visualize what the inverse transform should be, we expand the right hand side: PkkpU09MTl9sYXBHNiItSSdleHBhbmRHJSpwcm90ZWN0ZWRHNiRGI0kic0dGJA== Step 4: Now we have our algebraic solution "Y(s)" . We must find the inverse Laplace transform of 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. Note: If solving by hand we would first do a partial fraction expansion for 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. Once we have this we would use it to calculate the inverse Laplace transform of the second term and, in combination with Formula 1 from the table at the beginning of this worksheet, to compute the inverse Laplace transform of the third term. (Since the right hand side is not a rational expression we cannot just ask for a partial fraction from Maple on the entire expression.) : LUkoY29udmVydEclKnByb3RlY3RlZEc2JSomSSJzRzYiISIiLCZGJyIkKyYiIiQiIiJGKUkocGFyZnJhY0dGKEYn We instead compute the inverse Laplace transform using invlaplace -- making Maple do all of the work for computing the inverse. PkklU09MTkc2Ii1JK2ludmxhcGxhY2VHRiQ2JUkpU09MTl9sYXBHRiRJInNHRiRJInRHRiQ= LUknZXhwYW5kRyUqcHJvdGVjdGVkRzYjSSVTT0xORzYi As before we can plot our solution to see what the graph of the amount of salt in kg as a function of time t in minutes: PkknYW5zd2VyRzYiLUkkcmhzRyUqcHJvdGVjdGVkRzYjJkklU09MTkdGJDYjIiIi LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYnSSdhbnN3ZXJHRicvSSJ0R0YnOyIiISIjOi9JInlHRic7Ri0iJCsnL0kmY29sb3JHRidJJmdyZWVuR0YnL0kqdGhpY2tuZXNzR0YnIiIj LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=