How do you find the solution to the differential equation #dy/dx=cos(x)/y^2# where #y(π/2)=0#?

Question

Paisley Johnson · Accepted Answer

You can write:
#y^2dy=cos(x)dx# 
and integrate;
#inty^2dy=intcos(x)dx#
Which gives you:
#y^3/3=sin(x)+c#
#y^3=3sin(x)+c#
Now we find the value of #c#;
#(0)^3=3sin(pi/2)+c#
#0=3+c#
And #c=-3#
your particular solution is then:
#y^3=3sin(x)-3#

Adam Adkins · Answer

undefined To find the solution to the differential equation $\frac{dy}{dx} = \frac{\cos(x)}{y^2}$ with the initial condition $y(\frac{\pi}{2}) = 0$, follow these steps:

1. Separate the variables by multiplying both sides by $y^2$ and dividing both sides by $\cos(x)$, yielding $y^2 dy = \frac{1}{\cos(x)} dx$.

2. Integrate both sides with respect to their respective variables. This gives $\int y^2 dy = \int \sec(x) dx$.

3. Integrate $\int \sec(x) dx$ to get $\ln|\sec(x) + \tan(x)| + C_1$, where $C_1$ is the constant of integration.

4. Integrate $\int y^2 dy$ to get $\frac{1}{3}y^3 + C_2$, where $C_2$ is the constant of integration.

5. Combine the constants of integration into one constant, let's call it $C$, so the solution becomes $\frac{1}{3}y^3 + \ln|\sec(x) + \tan(x)| = C$.

6. Apply the initial condition $y(\frac{\pi}{2}) = 0$ to find the value of the constant $C$.

7. Substitute $x = \frac{\pi}{2}$ and $y = 0$ into the solution. This gives $0 + \ln|\sec(\frac{\pi}{2}) + \tan(\frac{\pi}{2})| = C$, which simplifies to $0 + \ln|1 + \infty| = C$.

8. Since $\tan(\frac{\pi}{2}) = \infty$, the absolute value of $\sec(\frac{\pi}{2}) + \tan(\frac{\pi}{2})$ is $\infty$, so $\ln|1 + \infty| = \infty$.

9. Therefore, the constant $C$ is $+\infty$.

10. The solution to the differential equation with the initial condition is $\frac{1}{3}y^3 + \ln|\sec(x) + \tan(x)| = +\infty$.