Elementary Differential Equations - …

Elementary Differential Equations First-order linear Differential Equations and solutions. Numerical solutions. Reference: E. Kreyszig,Advanced Engineering Mathematics, 5thed., Wiley, Linear Differential Equations General form takes:y +a(x)y=r(x),wherey =dy/dxandyis a function ofx. It is first order sincethe highest derivative is first order (y ). It is linear, because it is alinear function ofy andy(no terms likey2ory y, etc.). Homogeneous, whenr(x)=0. In this case, the solution can befound easily throughseparation of variables. Nonhomogeneous, whenr(x)6=0. The solution in this case isa bit more involved, and several different approaches 1st-order Linear Diff. Start withy +a(x)y= 02. Subtracta(x)yfrom both sides:y = a(x)y3. Divide both sides byy:1ydydx= a(x)4. Multiply both sides withdx:1ydy= a(x)dx5. Integrate:Z1ydy=Z a(x)dxlny= Za(x)dx+c6. Applyexp( ):y= exp Za(x)dx+c y= exp Za(x)dx exp(c)y=Cexp Za(x)dx This calledseparation of 1st-order Diff.

First-order Linear Differential Equations • General form takes: y0 + a(x)y = r(x), where y0 = dy/dx and y is a function of x.It is first order since the highest derivative is first order (y0).It is linear, because it is a

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  Differential, Equations, Elementary, Elementary differential equations, Differential equations

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