The first-order separable ordinary differential equations have the form of , allowing integration of both sides independently to find the general solutions.
Step 1, recognize the type of ODE, e.g., the 1st separable ordinary differential equations, and write it as
Step 2, take the integral on both sides,
Step 3, the integration on either side may require various standard techniques, i.e., simplify the LHS and RHS of Step 2
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Direct Integration & Basic Forms Intro. Read the following descriptions and select the last button to start
Many ODEs, once variables are separated, reduce to elementary integral forms. This is common when
the functions of x and y are simple polynomials, exponentials, or basic trigonometric functions that integrate directly
Examples include integrals leading to logarithms (e.g., ) or simple power rules.
Problem often involves in solving an integration constant using initial conditions (boundary values) particular solution
, given y(1) = 1/2.
D)
Find the general solution of
Solve the differential equation , given .
A quantity P grows at a rate proportional to its current value. If , and P(0) = 100, find P(t).
Find the general solution of for .
If and , find .
Solve , given y(0) = 1.
Find the general solution of .
The rate of change of temperature T of a cooling object is . If T(0) = 100, find T(t).
Solve the differential equation , given y(1) = e.
State the value of as becomes large ( ).