# Ordinary Differential Equations

Explain how to solve differential equations of the following types:

y'=f(ax+by+c)

y'=f(y/x)

x=f(y,y')

Provide examples with complete solutions for each case.

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I. The method. Let with f a continuous function. There is a simple trick to solve this type of differential equation: let us introduce the auxiliary function

(recall that y is a function of x). Then:

and So, by simple substitution, we get:

which is very easy to solve!

Example: to solve we put u=x+y+3, so we have to solve:

because a=b=1 in this case. The ...

#### Solution Summary

Ordinary differential equations are solved. The solution is detailed and well presented. Examples with completed solutions for each case are determined.

Solutions to First-Order Ordinary Differential Equations

Question 1: (1 points)

Solve the following equations for given initial condition.

Please represent your answer in the implicit form ( ). Use y(x) instead of y

Question 2: (1 points)

Integrate the following equations and write a solution for given initial condition.

Please represent your answer in the implicit form ( ). Use y(x) instead of y.

Question 3: (1 points)

Integrate the following equations and write a nontrivial solution.

Please represent your answer in the implicit form ( ). Use MAPLE syntax. Use y(x) instead of y.

Question 4: (1 points)

Integrate the following equations and write a nontrivial solution.

Please represent your answer in the implicit form ( ). Use MAPLE syntax. Use y(x) instead of y.

Question 5: (1 points)

Integrate the following equations and write a nontrivial solution.

Please represent your answer in the implicit form ( ). Use MAPLE syntax. Use y(x) instead of y.

Question 6: (1 points)

Integrate the following equations and write a nontrivial solution.

Please represent your answer in the implicit form ( ). Use MAPLE syntax.

Question 7: (1 points)

Integrate the following equations and write a solution for given initial condition.

Please represent your answer in the explicit form ( ). Use MAPLE syntax. Use y(x) instead of y.

Question 8: (1 points)

Integrate the following equations and write a nontrivial solution.

Please represent your answer in the explicit form ( ). Use MAPLE syntax.

Question 9: (1 points)

Solve the following equation

Please represent your answer in the implicit form ( ), Use MAPLE syntax. Use y(x) instead of y.

Question 10: (1 points)

Make the change of variable for the homogenous differential equation

and find , where

Question 11: (1 points)

Solve the following equation

Please represent your answer in the explicit form ( ), Use MAPLE syntax. Use y(x) instead of y.

Question 12: (1 points)

Find the general solution of the linear differential equation (use C to denote a constant)

.

Question 13: (1 points)

Find the solution of the linear differential equation which satisfies the initial condition :

.

Question 14: (1 points)

Find the solution of the following differential equation:

.

Question 15: (1 points)

Solve the following equation for given initial condition

Please represent your answer in the explicit form ( ), Use MAPLE syntax. Use y(x) instead of y.

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