Find the value of b for which the differential equation 5xy (yešey + x) dx + (bæe*v) dy = 0 is exact, then solve it using that value of b. The differential equation is exact only when The general solution, in implicit form, is C, for any constant c. NOTE: Do not enter an arbitrary constant or any constant term.
Find the value of b for which the differential equation 5xy (yešey + x) dx + (bæe*v) dy = 0 is exact, then solve it using that value of b. The differential equation is exact only when The general solution, in implicit form, is C, for any constant c. NOTE: Do not enter an arbitrary constant or any constant term.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Finding the Exact Value and General Solution of a Differential Equation
To determine the value of \( b \) for which the differential equation
\[ \left( ye^{5xy} + x \right) dx + \left( bxe^{5xy} \right) dy = 0 \]
is exact, and subsequently solve the equation using that value of \( b \).
#### Step 1: Determine the Value of \( b \)
The differential equation is exact only when
\[ b = \boxed{5} \]
#### Step 2: Solve the Differential Equation
The general solution, in implicit form, given the value of \( b \), is:
\[ \boxed{xe^{5xy} + \frac{y}{5} = c} = c, \quad \text{for any constant } c. \]
**Note:** Do not enter an arbitrary constant or any constant term.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56945fae-aeca-4f60-8c3d-a266c334f238%2F023d6c54-eb89-467d-aa20-ca2b0ded92a4%2Fgrd5xr4_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Exact Value and General Solution of a Differential Equation
To determine the value of \( b \) for which the differential equation
\[ \left( ye^{5xy} + x \right) dx + \left( bxe^{5xy} \right) dy = 0 \]
is exact, and subsequently solve the equation using that value of \( b \).
#### Step 1: Determine the Value of \( b \)
The differential equation is exact only when
\[ b = \boxed{5} \]
#### Step 2: Solve the Differential Equation
The general solution, in implicit form, given the value of \( b \), is:
\[ \boxed{xe^{5xy} + \frac{y}{5} = c} = c, \quad \text{for any constant } c. \]
**Note:** Do not enter an arbitrary constant or any constant term.
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