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# Solutions to First-Order Ordinary Differential Equations

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Question 1: (1 points)
Solve the following equations for given initial condition.

Question 2: (1 points)
Integrate the following equations and write a solution for given initial condition.

Question 3: (1 points)
Integrate the following equations and write a nontrivial solution.

Question 4: (1 points)
Integrate the following equations and write a nontrivial solution.

Question 5: (1 points)
Integrate the following equations and write a nontrivial solution.

Question 6: (1 points)
Integrate the following equations and write a nontrivial solution.

Question 7: (1 points)
Integrate the following equations and write a solution for given initial condition.

Question 8: (1 points)
Integrate the following equations and write a nontrivial solution.

Question 9: (1 points)
Solve the following equation

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

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

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#### Solution Preview

Question 1: (1 points)
Solve the following equations for given initial condition.

This is a separable differential equation. We write it as
.

Integrating both sides we obtain

for some constant c. Plugging in the initial condition we obtain

whence . Thus the solution is , or

The implicit form of this equation is .

Question 2: (1 points)
Integrate the following equations and write a solution for given initial condition.

Dividing both sides by we obtain

which is separable, so we write it as

.
Integrating both sides we obtain

.

Exponentiating both sides we now obtain

.

Plugging in the initial condition we obtain

.

Thus the solution is , or .

Question 3: (1 points)
Integrate the following equations and write a nontrivial solution.

This is another separable differential equation. Separating variable and integrating both sides we obtain
.
To evaluate the integral on the right (with the minus sign), we make the substitution , whence and the integral becomes

.

Similarly, the integral on the left becomes

.

Thus we have

whence

,

which yields a nontrivial solution for any constant c such that . For instance, when we have the nontrivial solution .

Question 4: (1 points)
Integrate the following equations and write a nontrivial solution.