MATH 20D Lecture 4

Integrate $\frac{dx}{(9x^2+4{)}^2}$

Let

Solve the following IVP: $x\frac{dy}{dx}=\frac{x^2+1}{(x+1{)}^2(y-1)},y(1)=2$

1. Separate:
2. Integrate:
3. Use initial condition: to solve for C:
   Implicit solution:
   
   
   
   
   
   so we take the + version
   Explicit solution:

Homogenous

If the RHS of the equation can be expressed as a function of the ratio only, then the equation is said to be homogenous. Such equations can always be transformed into separable equations by a change of dependent variable.

Solve the equation $\frac{dy}{dx}=\frac{y-4x}{x-y}$

Derive with respect to :

1. Separate :
2. Integrate
   
   (partial fraction decomp)
   
   
   
   
   
   
   
   General solution in implicit form:

Linear First-Order Equation

y: dependent variable, x: independent variable, y(x) unknown

Method for solving 1st-Order Linear Equations

Example: Find the general solution of

1. Convert the given into standard form (if needed). Then compute the integrating factor given by:
   
   
   
   So is our integrating factor
2. Multiply to both sides of the equation in standard form:
   
   
   Check if the derivative of is
   Convert LHS into a single derivative using product rule
   
3. Integrate both sides of then solve for
   
   
   Explicit form, general solution:
   
   Let
   
   So
   Now we use integration by parts on
   Let
   
   

Find the general solution of $(1+t^2)\frac{dy}{dt}+4ty=(1+t^2{)}^{-2}$

Convert the given equation into standard form by dividing both sides by



Multiply to both sides of the standard form eqn