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ALMATY UNIVERSITY OF POWER ENGINEERING AND TELECOMMUNICATIONS
Department of Higher Mathematics
Numerical methods and their computer realization
Laboratory work No.3
Problems of integral calculus, operational calculus, and
differential equations
Variant 11
Done by: the student of the group EPEe-16-8
Kabdygali S.M
Checked by: ass. prof. of the department of Higher mathematics
Kim R.E.
Almaty, 2017.
Content:
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Integrals;
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Operational calculus;
-
The calculation of differential equations with help of operational calculus and numerical methods of the MathCad program.
Task
1. Calculate the indefinite integral
.
Build three graphs from the family of antiderivatives.
Task performance.
Task
2. Calculate the definite integral
[1.5;2.5]
Task performance.
Task 3. Find the function image (Laplace image).
Task performance.
Task 4. Find the originals of the given functions (the inverse Laplace transform).
Task performance.
Task 5. Solve the differential equation with help of operational calculus (Laplace transform).
Task performance.
If we substitute values to the given differential equation, we will obtain the image of this equation:
Let’s solve it relative to the Y(s):
By applying to the resulting expression the inverse Laplace transform we find the original of Y(s):
Task 6. To solve the system of differential equations by the operational calculus.
x(0)
= -1, y(0) = 0
Task performance.
, 
Task 7.
1) Solve the first-order differential equation by the numerical method using the Given/Odesolve procedures of the Mathcad program.
2) Draw a graph of the solution obtained.
3) Find the value of the solution (function) obtained at the point x = 2.8.
Task performance.
Given
Answer: y(1.36) = -1.847*103, the graph is above.
Task 8.
1) Solve the second-order differential equation.
2) Draw a graph of the solution (function) obtained.
3) Create a table of arguments and function values.
4) Find the value of the solution obtained at t = 2.1 or select a point yourself.
Task performance.
Given
,
Answer: y(2.1) = 36.327, the graph and the table are above.
Task 9.
1) Solve the system of differential equations using the Given/Odesolve procedures. The initial conditions are the same for all variants of tasks: x1(0) = 2, x2(0) = 1.
2) Create a table of arguments and function values.
3) Find the value of the obtained functions at a point t = 3,4 or select a point yourself.
4) Build the graphs of the received functions.
5) Build a phase portrait of the system.
Task performance.
Given
,
-
the argument’s step is 0.1;
, 
Task 10.
1) Solve the system of differential equations (standalone) with the rkfixed function of the Mathcad program. The initial conditions are the same for all variants of the tasks: x1(0) = 3, x2(0) = 1.
2) Create a table of arguments and function values.
3) Find the value of the obtained functions at the point t = 4.2.
4) Build the graphs of the received functions.
5) Build a phase portrait of the system.
Task performance.
