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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Solution Preview
Please see the attachment.
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
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.
Please represent your answer in the implicit form ( ). Use y(x) instead of y.
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.
Please represent your answer in the implicit form ( ). Use MAPLE syntax. Use y(x) instead of y.
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.
Please represent your answer in the implicit form ( ). Use MAPLE syntax. Use y(x) instead of y.
This is another separable ...
Solution Summary
We solve several first-order ordinary differential equations. Implicit equations are examined.