91Ó°ÊÓ

#tag_title# Step 2: Multiply the Entire Equation by the Integrating Factor #tag_content# Now, we need to multiply both sides of the differential equation by the integrating factor, which is \(r(t) = e^{\frac{t^2}{2}}\): $$e^{\frac{t^2}{2}}y'(t) + te^{\frac{t^2}{2}}y^{1/3}(t) = e^{\frac{t^2}{2}} \tan(t)$$ #tag_title# Step 3: Integrate Both Sides #tag_content# Integrating both sides with respect to \(t\): $$\int \left[e^{\frac{t^2}{2}}y'(t) + te^{\frac{t^2}{2}}y^{1/3}(t)\right]dt = \int e^{\frac{t^2}{2}}\tan(t)dt$$ Let's use the substitution \(v(t)=y(t)e^{\frac{t^2}{2}}\), then \(v'(t)=y'(t)e^{\frac{t^2}{2}}+ty(t)e^{\frac{t^2}{2}}y^{\frac{1}{3}}(t)\). Substituting into the equation: $$\int v'(t)dt = \int e^{\frac{t^2}{2}}\tan(t)dt$$ We obtain: $$v(t) = \int e^{\frac{t^2}{2}}\tan(t)dt + C$$ #tag_title# Step 4: Solve for \(y(t)\) #tag_content# We found that \(v(t)=y(t)e^{\frac{t^2}{2}}\). So, to find \(y(t)\), we need to divide both sides by \(e^{\frac{t^2}{2}}\): $$y(t) = \frac{v(t)}{e^{\frac{t^2}{2}}} = \frac{\int e^{\frac{t^2}{2}}\tan(t)dt + C}{e^{\frac{t^2}{2}}}$$ #tag_title# Step 5: Apply the Initial Condition #tag_content# In order to find the unknown constant \(C\), we need to use the given initial condition \(y(-1) = 1\): $$1 = \frac{ \int e^{\frac{(-1)^2}{2}}\tan(-1)dt + C }{e^{\frac{(-1)^2}{2}}}$$ Solving for \(C\), we obtain: $$C = e^{-\frac{1}{2}} - \int e^{\frac{1}{2}}\tan(-1)dt$$ Substituting this value back into the expression for \(y(t)\), we obtain: $$y(t) = \frac{\int e^{\frac{t^2}{2}}\tan(t)dt + e^{-\frac{1}{2}} - \int e^{\frac{1}{2}}\tan(-1)dt}{e^{\frac{t^2}{2}}}$$