math270.mws First Order Differential Equations-Separation of Variables Example 1: Solve dy/dx=exp(y)*x/(exp(y)+x^2*exp(y). Example 2: Solve dy/dx-y=y*(x*exp(x)+1). Theorem: Example 3: Solve dy/dx=2*x^2+y-x^2*y+x*y-2*x-2. f(x,y)*f[xy]-f[x](x,y)*f[y](x,y) = 0; Example where f(x,y)*f[xy]-f[x](x,y)*f[y](x,y) = 0 fails: Linear First-Order Differential Equations Example 1: Solve dy/dx=y+x^2. Example 2: Solve dy/dx=x^3-2*x*y where y(1)=1. Exact Differential Equations Example: Solve the equation (1-sin(x)*tan(y))*dx+(cos(x)/(cos(y))^2)*dy=0. Example: Solve (y^2+2*y+1)*dx+(2*x*y+x^2+2)*dy=0. Example: Solve (3*x^2-y^2)*dy-2*x*y*dx=0. Second-Order Linear Differential Equation Example 1: Solve diff(y(x),x,x)+3*diff(y(x),x)-10*y(x)=0. Example 2: Solve diff(y(x),x,x)+6*diff(y(x),x)+9=0. Example 3: Solve diff(y(x),x,x)+diff(y(x),x)+y(x)=0. Example 4: Solve the differential equation from Example 3from Example 3, that satisfies the initial conditions y(0)=1, diff(y(0),x)=3. Variation of Parameters Method Example 1: Solve diff(y(x),x,x)-y(x)=exp(2*x) that satisfies initial conditions y(0)=0 and The method of undetermined coefficients Example 1: Find a particular solution of diff(y(x),x,x)+y(x)=x^2. Example 2: Find a particular solution of diff(y(x),x,x)-3*diff(y(x),x)+2*y(x)=exp(x)*sin(x). Example 3: Find a particular solution of diff(y(x),x,x)+y(x)=x*exp(2*x). Example 4: Find a particular solution of diff(y(x),x,x)+y(x)=cos(x). Example 5: Find a particular solution of diff(y(x),x,x)-4*diff(y(x),x)+4*y(x)=exp(2*x). Example 6: Find a particular solution of diff(y(x),x,x)+y(x)=x*sin(x). Euler Equations Example 1: Solve x^2*diff(y(x),x,x)+2*x*diff(y(x),x)-12*y(x)=0 . Example 2: Solve x^2*diff(y(x),x,x)-3*x*diff(y(x),x)+4*y(x)=0 . Example 3: Solve x^2*diff(y(x),x,x)+5*x*diff(y(x),x)+13*y(x)=0 . Example 4: Solve x^2*diff(y(x),x,x)+2*x*diff(y(x),x)-12*y(x)=sqrt(x) . Systems of First Order Linear Equations Example 1: Solve the system diff(x(t),t)=MATRIX([[-3, sqrt(2)], [sqrt(2), -2]])*x(t). Example 2: Solve the system diff(x(t),t)=MATRIX([[0, 1, 1], [1, 0, 1], [1, 1, 0]])*x(t) . In the case when eigenvalues are complex we have to find real and imaginary parts of the complex-valued solutions which we obtain in the same way as in the real case. Example 4: Solve the system diff(x(t),t)=MATRIX([[1,- 1], [1, 3]])*x(t) . Example 5: Solve the system diff(x(t),t)=MATRIX([[5,-3,-2], [8,-5,-4], [-4,3,3]])*x(t) .