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In this publication, we establish a suitable symmetry structure for orthogonal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>α</mi><mo>−</mo><mi mathvariant="fraktur">y</mi><mo>−</mo><mi mathvariant="double-struck">G</mi><mo>)</mo></mrow></semantics></math></inline-formula>-contractive mappings and prove fixed point results for an orthogonal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>α</mi><mo>−</mo><mi mathvariant="fraktur">y</mi><mo>−</mo><mi mathvariant="double-struck">G</mi><mo>)</mo></mrow></semantics></math></inline-formula>-contractive via orthogonal metric spaces. We give an application to strengthen our main results from the existing literature to prove the existence of a unique analytical solution to the differential equation by converting it into an integral equation through fixed point analysis.