Separation of Variables

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Separable Equations

Definition: Separable DE

If f(x,y) can be written as the product g(x)h(y), then the DE dydx=f(x,y) is separable.

Method

  1. Factor f(x,y) into g(x)h(y);

  2. Rewrite dydx=f(x,y) into dyh(y)=g(x)dx;

  3. Integrate both sides of equation: dyh(y)=g(x)dx;

  4. Solve for y, i.e., find an explicit solution, if possible. If it is not possible to solve for y, leave the answer as an implicit solution;

  5. If there is an initial condition, y(x0)=y0, use it to find the constant, C;

  6. Check for any missing solutions.

NOTE: If the solution to an ODE contains an integral that does not have an analytical antiderivative, then write the solution as an integral equation.

ODE Solution as Integral Equation

Example: Solve dydx=ex2y.

Answer:

Rewriting, we obtain

ydy=ex3dx

This implies

ydy=ex3dx

which implies

12y2=ex3dx

which implies

12y2=axet3dt

NOTE: There is no analytic antiderivative to ex3dx. Thus, the last line above is the solution to the ODE and is written as an integral equation using Fundamental Theorem of Calculus -- Part I.

Investigation 01

For each of the following, determine if the first-order ODE can be solved using separation of variables. Explain.

  1. dydx=3x2ey

  2. csc(x)dydx=x2y

  3. dydx=yx

  4. xsin2(y)dydx=x2+1

  5. y=e2x+y

  6. y=y+1x1

  7. y=xyx+1

  8. y=xy+2yx2xy3y+x3

  9. dydx=x2+y3

Investigation 02

Find the general solution to the following first-order ODEs using separation of variables [do not forget +C], if possible:

  1. dydx=3x2ey

  2. csc(x)dydx=x2y

  3. dydx=yx

  4. xsin2(y)dydx=x2+1

  5. y=e2x+y

  6. y=y+1x1

  7. y=xyx+1

  8. y=xy+2yx2xy3y+x3

  9. dydx=x2+y3

Investigation 03

Find the particular solution to the following first-order IVPs using separation of variables, if possible

  1. xdydx=y2+1, y(1)=1

  2. y=y2y2e3x, y(0)=1

  3. y=5xy2x, y(0)=1

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Last Modified: Monday, 31 August 2020 11:52 EDT