Exact Differential Equations Prepared by Vince Zaccone For

Exact Differential Equations Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

If a first-order DE can be written in the form: where M and N are both partial derivatives of the same function, the equation is EXACT This equation is EXACT if (these are partial derivatives) Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 1: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 1: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. These partials match, so the DE is exact. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 1: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. These partials match, so the DE is exact. To solve an exact equation, we need to integrate either M or N, realizing that the answers should only differ by a function of the other variable. Our result will be an implicit solution Ψ(x, y)=c. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 1: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. These partials match, so the DE is exact. To solve an exact equation, we need to integrate either M or N, realizing that the answers should only differ by a function of the other variable. Our result will be an implicit solution Ψ(x, y)=c. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 1: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. These partials match, so the DE is exact. To solve an exact equation, we need to integrate either M or N, realizing that the answers should only differ by a function of the other variable. Our result will be an implicit solution Ψ(x, y)=c. Instead of a constant, put an arbitrary function of y Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 1: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. These partials match, so the DE is exact. To solve an exact equation, we need to integrate either M or N, realizing that the answers should only differ by a function of the other variable. Our result will be an implicit solution Ψ(x, y)=c. Instead of a constant, put an arbitrary function of y Next we find the partial with y and equate it to the above expression to get a first-order DE for h(y). Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 1: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. These partials match, so the DE is exact. To solve an exact equation, we need to integrate either M or N, realizing that the answers should only differ by a function of the other variable. Our result will be an implicit solution Ψ(x, y)=c. Finally, we solve for h(y) and write down our solution. Instead of a constant, put an arbitrary function of y Next we find the partial with y and equate it to the above expression to get a first-order DE for h(y). Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 1: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. These partials match, so the DE is exact. To solve an exact equation, we need to integrate either M or N, realizing that the answers should only differ by a function of the other variable. Our result will be an implicit solution Ψ(x, y)=c. Finally, we solve for h(y) and write down our solution. Instead of a constant, put an arbitrary function of y Next we find the partial with y and equate it to the above expression to get a first-order DE for h(y). Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 1: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. These partials match, so the DE is exact. To solve an exact equation, we need to integrate either M or N, realizing that the answers should only differ by a function of the other variable. Our result will be an implicit solution Ψ(x, y)=c. Finally, we solve for h(y) and write down our solution. Instead of a constant, put an arbitrary function of y Next we find the partial with y and equate it to the above expression to get a first-order DE for h(y). Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

A quick overview of the method: A 1 st order DE that can be written as To find the solution we integrate either of these to get Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Integrating Factors Sometimes a DE can be made exact by multiplying through by some function. First put the equation in the form If there is an integrating factor that is a function of only one of the variables we can find it in the following way: In either case, multiply through the original equation by μ and the new DE should be exact. Then proceed as before. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 2: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 2: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. Not equal, so equation is not exact Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 2: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. Not equal, so equation is not exact Function of x only, so we can find int. factor Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 2: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. Not equal, so equation is not exact Function of x only, so we can find int. factor Solve this DE to find μ Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 2: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. Not equal, so equation is not exact Function of x only, so we can find int. factor Solve this DE to find μ Multiply through to get new equation Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 2: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. Not equal, so equation is not exact Function of x only, so we can find int. factor Solve this DE to find μ Multiply through to get new equation New equation – should be exact Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 2: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. Not equal, so equation is not exact Function of x only, so we can find int. factor Solve this DE to find μ Multiply through to get new equation New equation – should be exact These are equal, so equation is exact Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 2: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. New equation – should be exact Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 2: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. New equation – should be exact Now that we have an exact equation, we can solve it as before. Let’s integrate N (it looks simpler). Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 2: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. New equation – should be exact Now that we have an exact equation, we can solve it as before. Let’s integrate N (it looks simpler). Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 2: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. New equation – should be exact Now that we have an exact equation, we can solve it as before. Let’s integrate N (it looks simpler). Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 2: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. New equation – should be exact Now that we have an exact equation, we can solve it as before. Let’s integrate N (it looks simpler). Here is our implicit solution Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 3: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 3: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. Not equal, so equation is not exact Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 3: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. Not equal, so equation is not exact Not a function of x only, so we will try for the other variable. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 3: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. Not equal, so equation is not exact Not a function of x only, so we will try for the other variable. This is a function of y only, so we can find an integrating factor Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 3: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. Not equal, so equation is not exact Not a function of x only, so we will try for the other variable. This is a function of y only, so we can find an integrating factor This DE is separable – solve it for μ(y) Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 3: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. Not equal, so equation is not exact Not a function of x only, so we will try for the other variable. This is a function of y only, so we can find an integrating factor This DE is separable – solve it for μ(y) Multiplying through by μ(y) should make our equation exact. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 3: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 3: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. This equation should be exact Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 3: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. This equation should be exact To solve this one, let’s integrate M (simpler) Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 3: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. This equation should be exact To solve this one, let’s integrate M (simpler) Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 3: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. This equation should be exact To solve this one, let’s integrate M (simpler) Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 3: Is the following DE exact? If so, solve it. If not, try to find an integrating factor that will make it exact. This equation should be exact To solve this one, let’s integrate M (simpler) Here is our implicit solution Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB