Solving Differential Equations by Exact Differential Equations

Solving Differential Equations by Exact Differential Equations:

  1. Method Overview:

    • Applicable to first-order ordinary differential equations (ODEs) in the form M(x,y)dx+N(x,y)dy=0.
    • Checks for the existence of an integrating factor to make the equation exact, i.e., My=Nx.
  2. Steps to Solve by Exact Differential Equations:

    a. Check for Exactness: Given a differential equation in the form M(x,y)dx+N(x,y)dy=0, verify if My=Nx.

    b. Find Integrating Factor: If the equation is not exact, find a suitable integrating factor μ(x,y) such that after multiplication by this factor, the equation becomes exact. The integrating factor μ(x,y) is obtained by μ=eMyNxNdx.

    c. Multiply and Solve: Multiply the given equation by the integrating factor μ(x,y) and rearrange to obtain an equation that is now exact. Then, solve it using integration techniques.

  3. Example: Solving a Differential Equation by Exact Differential Equations:

    Consider the equation (2xy+1)dx+(x2+2y)dy=0.

    a. Check for Exactness:

    • Calculate My=2x and Nx=2x. As they are equal, the equation is exact.

    b. Solve the Exact Equation:

    • Integrate M(x,y)dx to find the function f(x,y): f(x,y)=(2xy+1)dx=x2y+x+g(y)
    • Take the partial derivative of f(x,y) with respect to y and set it equal to N(x,y) to find g(y): fy=x2+g(y)=x2+2y.
    • Solve for g(y) to get g(y)=y2+C.

    c. Final Result:

    • The solution to the differential equation (2xy+1)dx+(x2+2y)dy=0 by the method of exact differential equations is x2y+x+y2+C=0, where C is the constant of integration.

Example: Solving a Non-Exact Differential Equation by Integrating Factor

Consider the non-exact differential equation yyx=ex.

  1. Identify the Equation:

    • The given equation is non-exact since MyNx.
  2. Integrating Factor Method:

    • To make the equation exact, introduce an integrating factor μ(x) by multiplying the entire equation.
  3. Finding the Integrating Factor:

    • For the equation yyx=ex, the integrating factor is μ(x)=e1xdx=elnx=1x.
  4. Multiply by Integrating Factor:

    • Multiply the given equation yyx=ex by the integrating factor 1x to make it exact.

    1x(yyx)=1xex  1xy1x2y=1xex

  5. Solving the Exact Equation:

    • Now, solve the transformed exact equation.
    • Let 1xy1x2y=1xex be the exact equation.

    a. Integrate with Respect to x:

    • Integrate both sides of the equation with respect to x.
    • 1xydx1x2ydx=1xexdx
    • lnxy+1xy=ex+C, where C is the constant of integration.
  6. Final Result:

    • The solution to the non-exact differential equation yyx=ex using the integrating factor method is y=xex+Cx, where C is the constant of integration.