The differential equation $$ \frac{d y}{d x}=\cos (x) \frac{y^{2}+12 y+35}{9 y+55} $$ has an implicit general solution of the form $$ F(x, y)=K $$, where $$ K $$ is an arbitrary constant. In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form $$ F(x, y)=G(x)+H(y)=K . $$ Find such a solution and then give the related functions requested. $$ F(x, y)=G(x)+H(y)= $$

Question

UpStudy AI · Accepted Answer

We start with the differential equation
$$
\frac{dy}{dx}=\cos(x)\frac{y^{2}+12y+35}{9y+55}.
$$
Since the equation is separable, we first write it in the form
$$
\frac{9y+55}{y^{2}+12y+35}\,dy=\cos(x)\,dx.
$$

### Step 1. Factor and Decompose

Notice that the quadratic factors as
$$
y^{2}+12y+35=(y+5)(y+7).
$$
We then have
$$
\frac{9y+55}{(y+5)(y+7)}\,dy=\cos(x)\,dx.
$$

We now express the left-hand side as a sum of partial fractions. Assume
$$
\frac{9y+55}{(y+5)(y+7)}=\frac{A}{y+5}+\frac{B}{y+7}.
$$
Multiplying both sides by $$(y+5)(y+7)$$ gives
$$
9y+55 = A(y+7) + B(y+5).
$$
Expanding the right side:
$$
9y+55 = (A+B)y + (7A+5B).
$$
Thus, equating coefficients we have:
$$
A+B = 9 \quad \text{and} \quad 7A+5B = 55.
$$

Solve for $$A$$ and $$B$$:
- From $$A+B=9$$, we get $$B=9-A$$.
- Substitute into $$7A+5(9-A)=55$$:
  $$
  7A+45-5A =55 \quad\Longrightarrow\quad 2A=10 \quad\Longrightarrow\quad A=5.
  $$
- Then $$B=9-5=4$$.

Thus, we have:
$$
\frac{9y+55}{(y+5)(y+7)} = \frac{5}{y+5}+\frac{4}{y+7}.
$$

### Step 2. Integrate Both Sides

The separated equation is now:
$$
\left(\frac{5}{y+5}+\frac{4}{y+7}\right)dy = \cos(x)\,dx.
$$

Integrate both sides:
$$
\int \left(\frac{5}{y+5}+\frac{4}{y+7}\right)dy = \int \cos(x)\,dx.
$$

Compute the integrals:
- For the $$y$$-side:
  $$
  \int \frac{5}{y+5}\,dy = 5\ln|y+5|,\qquad \int \frac{4}{y+7}\,dy = 4\ln|y+7|.
  $$
  So the left-hand side becomes:
  $$
  5\ln|y+5| + 4\ln|y+7|.
  $$
- For the $$x$$-side:
  $$
  \int \cos(x)\,dx = \sin(x).
  $$

Thus, we have:
$$
5\ln|y+5| + 4\ln|y+7| = \sin(x) + C,
$$
where $$C$$ is an arbitrary constant.

### Step 3. Write the Final Answer in the Requested Form

We wish to express the solution as
$$
F(x,y) = G(x) + H(y) = K,
$$
where $$K$$ is a constant. We can rearrange the integrated equation to obtain:
$$
\sin(x) - 5\ln|y+5| - 4\ln|y+7| = K,
$$
where $$K = -C$$ (since $$C$$ is an arbitrary constant, so is $$K$$).

Thus, we define:
$$
G(x)=\sin(x), \quad H(y)=-5\ln|y+5|-4\ln|y+7|,
$$
and so the implicit solution is:
$$
F(x,y)=\sin(x)-5\ln|y+5|-4\ln|y+7|=K.
$$
$$ F(x, y) = \sin(x) - 5\ln|y+5| - 4\ln|y+7| = K $$

Upstudy AI Solution

Answer

Solution

Step 1. Factor and Decompose

Step 2. Integrate Both Sides

Step 3. Write the Final Answer in the Requested Form

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The differential equation has an implicit general solution of the form , where is an arbitrary constant. In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form Find such a solution and then give the related functions requested.