Answer:
Verified
Step-by-step explanation:
Question:-
- We are given the following non-homogeneous ODE as follows:
- A general solution to the above ODE is also given as:
- We are to prove that every member of the family of curves defined by the above given function ( y ) is indeed a solution to the given ODE.
Solution:-
- To determine the validity of the solution we will first compute the first derivative of the given function ( y ) as follows. Apply the quotient rule.
- Now we will plug in the evaluated first derivative ( y' ) and function ( y ) into the given ODE and prove that right hand side is equal to the left hand side of the equality as follows:
- The equality holds true for all values of " C "; hence, the function ( y ) is the general solution to the given ODE.
- To determine the complete solution subjected to the initial conditions y (1) = 3. We would need the evaluate the value of constant ( C ) such that the solution ( y ) is satisfied as follows:
- Therefore, the complete solution to the given ODE can be expressed as:
- To determine the complete solution subjected to the initial conditions y (3) = 1. We would need the evaluate the value of constant ( C ) such that the solution ( y ) is satisfied as follows:
- Therefore, the complete solution to the given ODE can be expressed as:
8: slope is 1/2 and y intercept is 0
9: slope is 7/4 and y-intercept is 0
Answer:
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Step-by-step explanation:
Answer:
<em>Given </em><em>points </em><em>are </em><em>(</em><em> </em><em>-</em><em>2</em><em> </em><em>,</em><em> </em><em>-</em><em>5</em><em> </em><em>)</em><em> </em><em>and </em><em>slope </em><em>(</em><em> </em><em>m</em><em>) </em><em> </em><em>=</em><em> </em><em>-</em><em>4</em>
Step-by-step explanation:
<em>y </em><em>-</em><em> </em><em>y1</em><em> </em><em>=</em><em> </em><em>m </em><em>(</em><em> </em><em>x </em><em>-</em><em> </em><em>x1</em><em>)</em>
<em>y </em><em>+</em><em> </em><em>5</em><em> </em><em>=</em><em> </em><em>-</em><em>4</em><em> </em><em>(</em><em> </em><em>x </em><em>+</em><em> </em><em>2</em><em> </em><em>)</em>
<em>y </em><em>+</em><em> </em><em>5</em><em> </em><em>=</em><em> </em><em>-</em><em>4</em><em>x</em><em> </em><em>-</em><em> </em><em>8</em>
<em>4x </em><em>+</em><em> </em><em>y </em><em>+</em><em>5</em><em> </em><em>+</em><em> </em><em>8</em><em> </em><em>=</em><em> </em><em>0</em>
<em>4x </em><em>+</em><em> </em><em>y </em><em>+</em><em>1</em><em>3</em><em> </em><em>=</em><em> </em><em>0</em><em> </em>
<em>y </em><em>=</em><em> </em><em>-</em><em>4</em><em>x</em><em> </em><em>-</em><em> </em><em>1</em><em>3</em>
<em>which </em><em>is </em><em>the </em><em>required </em><em>equation </em>