Solve differential equation: y' + y^2 sinx = 0

abondantQ

abondantQ

Answered question

2021-03-05

Solve differential equation: y+y2sinx=0

Answer & Explanation

timbalemX

timbalemX

Skilled2021-03-06Added 108 answers

dydx=y2.sin(x)
dyy2=sin(x)dx
1y=C+cos(x)
y(x)=1C+cos(x)
Mr Solver

Mr Solver

Skilled2023-06-19Added 147 answers

Step 1: Separation of Variables
We start by rearranging the equation to isolate the variables on opposite sides:
y=y2sinx.
Now, we can separate the variables by dividing both sides by y2:
dyy2=sinxdx.
Step 2: Integration
We integrate both sides of the equation with respect to their respective variables. On the left side, we integrate with respect to y, and on the right side, we integrate with respect to x:
dyy2=sinxdx.
Step 3: Evaluating the Integrals
The integral of dyy2 can be evaluated using the power rule of integration. The integral of sinx is a straightforward integral. Applying these integrals, we get:
1y=cosx+C,
where C is the constant of integration.
Step 4: Solving for y
To solve for y, we can multiply both sides of the equation by 1 to get rid of the negative sign:
1y=cosxC.
Next, we take the reciprocal of both sides to isolate y:
y=1cosxC.
Step 5: Final Solution
The final solution to the given differential equation is:
y(x)=1cosxC.
Eliza Beth13

Eliza Beth13

Skilled2023-06-19Added 130 answers

The given differential equation is:
dydx+y2sin(x)=0
To solve this equation, we can use the method of separation of variables. Rearranging the equation, we have:
dyy2=sin(x)dx
Integrating both sides, we get:
dyy2=sin(x)dx
Simplifying the integrals, we have:
1y=cos(x)+C
To find the value of the constant, we can use an initial condition. Let's assume y(0)=y0, where y0 is the initial value. Substituting x=0 and y=y0 into the equation, we get:
1y0=cos(0)+C
Simplifying further, we have:
1y0=1+C
Solving for C, we find:
C=11y0
Substituting this value back into the equation, we get the final solution:
1y=cos(x)11y0
Therefore, the general solution to the given differential equation is:
y(x)=1cos(x)11y0
madeleinejames20

madeleinejames20

Skilled2023-06-19Added 165 answers

Answer:
y=1cos(x)+C
Explanation:
dydx=y2sin(x)
Now, let's separate the variables and integrate both sides:
1y2dy=sin(x)dx
Integrating the left side yields:
1y=sin(x)dx
The integral on the right side can be evaluated as:
1y=cos(x)+C where C is the constant of integration. Next, we can solve for y by taking the reciprocal:
y=1cos(x)+C
Thus, the general solution to the given differential equation is:
y=1cos(x)+C where C is an arbitrary constant.

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